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COPYRIGHT DEPOSIT. 



AMERICAN MACHINIST GEAR BOOK 



3 



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AMEEICAN MACHINIST 
GEAR BOOK 



Simplified Tables and Formulas 
for Designing, and Practical Points in 
Cutting All Commercial Types of Gears 



BY 

CHARLES H. LOGUE 

FORMERLY ASSOCIATE EDITOR AMERICAN MACHINIST 
FORMERLY MECHANICAL ENGINEER R. D, NUTTALL CO. 

Thoroughly Revised by 

REGINALD TRAUTSCHOLD, M. E. 



Third Edition 



McGRAW-HILL BOOK COMPANY, Inc. 
NEW YORK: 370 SEVENTH AVENUE 

LONDON: 6 & 8 BOUVERIE ST., E. C. 4 



WJ^ 






Copyright, 1922, by the 
McGraw-Hill Book Company, Inc. 

Copyright, 1918, by the 
McGraw-Hill Book Company, Inc. 



Copyright, 1910, by the American Machinist 
Copyright, 1915, by the American Machinist 



^^~^H^> 



THE MAPLE PRESS - YORK PA 



DEC -9 72 

)CUe92308 



PREFACE TO THIRD EDITION 

The continued demand for a work on gearing along lines as originally 
planned by Charles H. Logue in 1910 has been so persistent that it has been 
deemed advisable to enlarge and revise the previous edition of the American 
Machinist Gear Book rather than to publish at the present time a new book 
on practical gear design and methods of production. In doing so, parts of 
the original work which are no longer of particular value to the practical man 
have been omitted and other parts revised to bring the work up to date. The 
chief additions have consisted of sections devoted to varieties of bevel gears 
which have demonstrated their commercial value during the past few years — 
spiral type bevel gears and bevels of the Williams "master form" variety; to 
an exposition of the Williams System of Internal Gearing; and to the first 
presentation in book form of the successful development of the rolling process 
of gear production. 

The section devoted to " Costs" in the Revised Edition has been omitted, 
not so much because it has grown obsolete, but because the developments in 
production methods have been so marked and radical in recent years that the 
question of costs has been complicated by a variety of production processes 
which do not lend themselves to direct comparison. The section on "Prac- 
tical Points in Gear Cutting" has also been omitted for the reason that gear 
cutting has now been very largely superseded by gear generation and the 
helpful hints on gear cutting are of little value in modern processes of gear 
generation. 

Reginald Trautschold, 
New York, N. Y. 

September, 1922, 



PREFACE TO FIRST EDITION 

This book has been written to fill a pressing want; to give practical data 
for cutting, molding, and designing all commercial types, and to present these 
subjects in the plainest possible manner by the use of simple rules, diagrams, 
and tables arranged for ready reference. In other words, to make it a book 
for "the man behind the machine," who, when he desires information on a 
subject, wants it accurate and wants it quick, without dropping his work to 
make a general study of the subject. At the same time a general outline of 
the underlying principles is given for the student, who desires to know not 
only how it is made, but what is made. Controversies and doubtful theories 
are avoided. Tables and formulas commonly accepted are given without 
comment. A great deal of this matter has previously been published in the 
columns of the American Machinist, but is revised to make the subject 
more complete. Credit is given in all cases when the author is known; there 
may be cases however, where record of the original source of information has 
been lost, as is often the case when data are in daily use and the authority 
is obscure. Obviously in such cases the author's name cannot be given. 

Charles H. Logue. 
May I, 1910. 

PREFACE TO SECOND EDITION 

The first edition of this work, published some five years ago, met with 
gratifying success as it filled a widespread demand for reliable information 
on the subject of gearing. The last few years have added much to our 
knowledge concerning certain types of gears however, so that the present 
is a fitting time to revise and bring this work up to date. In doing so, the 
original plan of the work has been followed and much of its text retained 
intact, while considerable new matter has been added. The sections on 
bevel, worm, helical, and skew bevel gears have been rewritten. The 
section on pattern work and molding has been omitted and one touching 
upon the cost of standard gears substituted therefor. 

Reginald Trautschold. 
September i, 1915. 



vu 



CONTENTS 

Preface to Third Edition v 

Preface to Second Edition vii 

Preface to First Edition vii 

Section Page 

I. Tooth Parts i 

II, Spur Gear Calculations 42 

III. Speeds and Powers 46 

IV. Gear Proportions and Details of Design 102 

V. Bevel Gears 130 

VI. Worm Gears 152 

VII. Helical and Herringbone Gears 190 

VIII. Spiral Gears 210 

IX. Skew Bevel Gears 228 

X. Intermittent Gears 236 

XL Elliptical Gears 246 

XII. Epiclycic Gear Trains 253 

XIII. Friction Gears 268 

XIV. Special Bevel Gears 303 

XV. Williams System of Internal Gearing 320 

XVI. Rolled Gearing 334 

Index 349 



JX 



AMERICAN MACHINIST 
GEAR BOOK 

SECTION I 

Tooth gearing furnishes an efficient and simple means of transmitting power 
at a constant speed ratio, making it possible to time the movements of 
machine parts positively. 

Owing to refinements in the tooth form, the introduction of generating 
machines and facilities to cut gears of the largest size accurately, loads may 
now be transmitted at speeds which a comparatively short time ago were 
considered prohibitive. The need for gears that would answer the exacting 
requirements of automobile construction has done much to bring this about. 
Designing automobile gears, however, is a case of fitting the gears to the 
machine; it is a question of securing material that will stand the strain; the 
gear dimensions are practically self determined; however, this is not the only 
kind of gearing that has been designed after this fashion. 

We have an excellent formula for the strength of gear teeth, but it contains 
a variable factor — the allowance to be made on account of impact — concern- 
ing which very little is known. The most important question of all, that of 
wear, has heretofore been left practically untouched. The best data ob- 
tainable has been given. Few records have been kept of actual performances, 
and nothing whatever has been found relative to the abrasion of different 
materials in tooth contact. 

The various ways in which gears are mounted is responsible for the appa- 
rent contradiction of what few data are at hand, as a gear driver which is 
entirely satisfactory on one machine will be worthless on another at the same 
load and the same speed. 

The circumferential speed that may be allowed for gears of different types 
is another neglected factor, and last, but not least, what do we know of gear 
efficiency? In fact the most important information relative to gear trans- 
missions has been entirely a matter of guesswork. It is hardly to be assumed 
that this will ever be reduced to an exact equation, but there should be some 
basis from which to form our conclusions. 

Gears may be roughly divided into three general classes : Gears connecting 
parallel shafts ; gears connecting shafts at any angle in the same plane ; gears 
connecting shafts at any angle not in the same plane. 

In the first class are included spur, helical, herringbone, and internal gears. 
The second class covers bevel gears only. The third class includes worm^ 
spiral and skew bevel gears. 



2 AMERICAN MACHINIST GEAR BOOK 

Gears connecting parallel shafts are the most efficient, and from a point of 
efficiency may be graded into herringbone, internal, spur, and, lastly, helical 
gears. 

The efficiency of the second class, bevel gears, varies with the shaft angle, 
increasing as the angle approaches zero. 

As a general thing the third class should be avoided wherever possible, 
although worm gears have their peculiar uses; for instance, where a quiet, 
self-locking drive is required without reference to the loss of power. 

Spiral gears are employed where the load is light and the gear ratio is low, 
say under lo to i; worm gears are often employed for low ratios down to i 
to I, but are extremely difficult to cut and therefore expensive. When the 
worm is made much coarser than quadruple thread there is generally trouble. 

Skew bevel gears are used where the distance between the shafts is not 
great enough to employ worm or spiral gears. Skew bevel gears are simpler, 
and easier to cut than has been generally supposed, but are still things to 
avoid. 

These three general classifications are commercially subdivided as follows: 



KIND 


RELATION or AXES 


PITCH SURFACES 


NOTES 


Spur 
Bevel 


Parallel 
Intersecting at any angle in 
the same plane 


Cylinders 

Cones 

Cylinders 

Cylinders 

Cylinders 

Cylinders 

Hyperboloids 

Cylinders or cones 

Elliptical cylinders or 
elliptical cones 
Any 

Cylinders 
Cyhnders or cones 




Helical 

Herringbone 

Spiral 


ParaUel 
Parallel 
At any angle not in same 
plane 


Double Helical 

For small ratios 

For large ratios 

Where shaft centers 
are close 


Worm 


At any angle not in same 
plane 


Skew Bevel 


At any angle not in same 
plane 


Internal 


Parallel or at any angle in 
same plane 


Elliptical 


Parallel or at any angle in 
same plane 


ner surface 


Irregular 


Parallel or at right angles in 
same plane 


Irregular pitch lines 
To give driven gear a 
period, or periods of 
rest during one rev- 
olution of driver 
Contact surfaces rep- 
resenting the pitch 
surfaces of a toothed 
gear 


Intermittent 


Parallel or at right angles in 
same plane 


Friction 


Parallel or at any angle in 
same plane 







Commercial Classification of Gears 



TOOTH PARTS 

Fixed axes are connected by imaginary pitch surfaces, which roll upon each 
other and transmit uniform motion without slipping. The object in toothed 
gearing is to provide these imaginary surfaces with teeth, the action of which 



TOOTH PARTS 



will make the uniform motion of the pitch surfaces positive ; not depending 
upon friction produced by lateral pressure as in friction gears, which are an 
excellent representation of pitch surfaces. 

If the teeth are not so formed that this condition is fulfilled the movement 
of the driven gear will be made up of accelerations and retardations which 
will not only absorb a large percentage of the power but disintegrate the 
material of which the gear is constructed and seriously affect the operation of 
the machine. Tool marks on planer and 
boring mill work corresponding to the teeth 
in the driven gear may be traced directly 
to this. 

There is but one form of tooth in com- 
mon use — the involute; the cycloidal form 
has practically disappeared. For a 
thorough understanding of tooth contact, 
however, it must be included. 

CYCLOIDAL 




FIG. I. 



Generated by rolling a circle above 
and below the pitch circle of gear; a point 
on its circumference describing the tooth outline 



GENERATING THE CYCLOIDAL 
TOOTH. 



See Fig. i. 



INVOLUTE 



Generated by rolling a straight line on the base circle of gear, any point on 
this line describing the involute curve. See Fig. 2. The same result is 
obtained by unwinding a string from the base circle. See Fig. 3. 



OCTOID 

Conjugated by a tool representing a flat sided crown gear tooth; a modi- 
fication of the involute. Used only on bevel gear generating machines. See 
Fig. 33- 

THE CYCLOID 

An illustration of the manner in which the cycloidal tooth is generated is 
illustrated by Fig. i; the wheel A being the pitch circle and B and B' the 
describing circles which are of the same diameter. The point C will describe 
the face of the tooth as the circle B is rolled on the pitch circle, and the flank 
of the tooth as the circle B' is rolled inside the pitch circle. In other words, 
the exterior cycloid is formed by rolling the describing circle on the outside 
of the pitch circle, this exterior cycloid engaging the interior cycloid, which 
is formed by rolling the describing circle on the inside of the pitch circle. 

The describing circle is commonly made equal to the pitch radius of a 15- 
tooth pinion of the same pitch as the gear being drawn. 



AMERICAN MACHINIST GEAR BOOK 



According to J. Howard Cromwell: " Roomer, a celebrated Danish astrono- 
mer, is said to have been the first to demonstrate the value of these curves for 
tooth profiles." But De la Hire is credited with demonstrating that it was 
possible to form both the face and flanks of any number of gears with the 
same describing circle. 

The pressure angle of the teeth is not constant in one direction, but varies 
from zero at the pitch point to about 22 degrees at the end of the contact 
with a rack tooth. The contact points of all the teeth engaged intersect the 
line of action, which is a segment of the describing circle drawn from the line 
of centers. See Fig. 34. 

Wilfred Lewis has said: ''The practical consideration of cost demands the 
formation of gear teeth upon some interchangeable system. 




Base Circle 



FIG. 2. THE INVOLUTE GEN- 
ERATED BY A STRAIGHT LINE. 




FIG. 3. THE INVOLUTE GEN- 
ERATED BY A STRING. 



*'The cycloidal system cannot compete with the involute, because its 
cutters are formed with greater difficulty and less accuracy, and a further 
expense is entailed by the necessity for more accurate center distances. 
Cycloidal teeth must not only be accurately spaced and shaped but their wheel 
centers must be fixed with equal care to obtain satisfactory results. Cut 
gears are not only more expensive in this system, but also when patterns 
are made for castings the double curved faces require far more time and care 
in chiseling. An involute tooth can be shaped with a straight-edged tool, 
such as a chisel or a plane, while the flanks of cycloidal teeth require special 
tools, approximating in curvature the outline desired. It is, therefore, hardly 
necessary to argue any further against the cycloidal gear teeth, which 
have been declining in popularity for many years, and the question now 
to be considered is the angle of obliquity most desirable for interchangeable 
involute teeth." 

In this same connection George B. Grant, of the Philadelphia Gear Works, 
wrote: ''There is no more need of two different kinds of tooth curves for 



TOOTH PARTS 5 

gears of the same pitch than there is need of two different threads for standard 
screws, or of two different coins of the same value, and the cycloidal tooth 
would never be missed if it were dropped altogether. But it was first in the 
field, is simple in theory, is easily drawn, has the recommendation of many 
well-meaning teachers and holds its position by means of 'human inertia,' 
or the natural reluctance of the average human mind to adopt a change, 
particularly a change for the better." 



THE INVOLUTE 

The pressure on the teeth of involute gears is constantly in the direction 
of the line of action. The line of action is drawn through the pitch point at an 
angle from the horizontal equal to the 
angle of obliquity. All contact be- 
tween the teeth is along this line. 
The base circle is drawn inside the 



Pitch Circle 





FIG. 4. THE ACTION OF INVOLUTE TEETH 
ILLUSTRATED BY A CROSSED BELT CON- 
NECTING THE BASE CIRCLES. 



FIG. 5. SEPARATING THE PITCH CIRCLES 
TO ALLOW THE EXTERIOR CYCLOIDS TO 
ENGAGE. 



pitch circle and tangent to the line of action. 

The action of a pair of involute gears is the same as if their base circles were 
connected by a cross belt; the point at which the belt crosses being the pitch 
point P; the straight portion of the belt not touching the base circles reore- 



6 AMERICAN MACHINIST GEAR BOOK 

senting the lines of action. See Fig. 4. At the pitch point the velocities of 
both gears are equal. To show that the involute is but a limiting case of the 
cycloidal system, consider the describing line as a curve of infinite radius, 
which is rolled upon the pitch circle. As this describing line cannot be rolled 
inside the pitch circle to form the interior cycloid that will engage the exterior 
cycloid formed by rolling the describing line outside the pitch circle of the 
mating gear, the pitch circles upon which the cycloids are formed must be 
separated so as to allow the exterior cycloids to engage each other. The orig- 
inal pitch circles becoming the base circles. See Fig. 5. 

The distance between the pitch circle and the base circle, and therefore, 
the angle of obliquity, depends upon the proportionate length of tooth to be 
used and the smallest number of teeth in the system. To obtain contact for 
the full length of the tooth, the base circle must fall below the lowest point 
reached by the teeth of the mating gear. Below the base line there can be no 
contact of any value. 

There is such a difference between the largest possible gear and the rack 
that it is at first a little difficult to see the application of the methods used to 
describe the involute to the rack tooth. As the diameter of the gear is in- 
creased, the radii used to draw the involute curve are lengthened, and the 
teeth have less curvature. Until finally, when the radius of the pitch circle 
is of infinite length, the tooth radii are also infinite, and the involute is a 
straight line, drawn at right angles to the line of action. 

The theoretical rack tooth, therefore, has perfectly flat sides, each side 
being inclined toward the center of the tooth to an angle equaling the angle of 
obHquity. See Fig. 6. 

ORIGIN OF THE INVOLUTE TOOTH 

The origin of the involute curve as applied to the teeth of gears is credited 
to De la Hire, a French scientist, a complete description and explanation of 
its use being published about 1694 in Paris. The first English translation of 
this work was published in London in 1696 by Mandy.* Professor Robinson, 
of Edinburgh, later describes this theory, references being made to his work 
in "An Essay on Teeth of Wheels," by Robertson Buchannan, edited by 
Peter Nicholson and published in 1808. In this essay the involute as 
applied to the teeth of gears is fully described. Fig. 7 being a copy of a cut 
used therein for illustration. That the principal advantage of the involute 
system was then well understood will be shown in the following paragraph, 
referring to Fig. 7: 

'' It is obvious that these teeth will work both before and after passing the 

* However, the origin of the involute gear tooth is surrounded by mystery, no two 
authorities agreeing upon the subject. According to Robert Willis, in his "Principles of 
Mechanism," the involute was first suggested for this purpose by Euler, in his second paper 
on the Teeth of Wheels. N. C. Petr XI. 209. 



TOOTH PARTS 



line of centers, they will work with equal truth, whether pitched deep or shal- 
low, a quality peculiar to them and of very great importance." 

The theory of the involute gear tooth is also described by Sir David Brew- 
ster, Dr. Thomas Young, Mr. Thomas Reid and others. 

Professor Robert Willis, gives a 
very complete description of this 
form of tooth in his "Principles of 
Mechanism," 1841. Up to this 
period the involute tooth was not 





FIG. 6. THE INVOLUTE RACK TOOTH. 



FIG. 7. ACTION OF THE INVOLUTE TOOTH. 



seriously considered, the cycloidal being the favorite. The involute tooth 
was objected to on account of the great thrust supposed to be put on the 
bearings by the oblique action of the teeth. 

In an 1842 edition of M. Camus' work, "A Treatise on the Teeth of 
Wheels," edited by John I. Hawkins, a series of experiments with wooden 
models was made to demonstrate the 
actual thrust occasioned by different 
angles of obliquity. The result of these 
experiments is given as follows: 

"These experiments, tried with the 
most scrupulous attention to every cir- 
cumstance that might affect their result, 
elicit this important fact — that the teeth 
of wheels in which the tangent of the sur- 
faces in contact makes a less angle than 20 degrees with the line of centers, 
possess no tendency to cause a separation of their axes: Consequently, there 
can be no strain thrown upon the bearings by such an obliquity of tooth." 

J. Howard Cromwell, in his treatise on Tooth Gearing, 1901, says: 
"Such an obliquity as 20 degrees must, unless counteracted by an opposing 
force, tend to separate the axes; and, as suggested by Mr. Hawkins, this 
opposing force is most probably the friction between the teeth, which would 




FIG. 8. THE MOLDING PROCESS. 



8 



AMERICAN MACHINIST GEAR BOOK 



tend to drag the axes together with as much force as that tending to separate 
them." 

That the involute system is closely connected to the cycloidal system is 
shown by Dr. Brewster in his reference to De la Hire's work. 

''De la Hire considered the involute of a circle as the last of the exterior 
epicycloids; which it may be proved to be, if we consider the generating 
straight line (see Fig. 2) as a curve of infinite radius." 

The 143^^ degree angle of obliquity, as proposed by Professor Robert Willis 
in his "Principles of Mechanism," was adopted by the Brown & Sharpe 
Company some forty years ago. Since that time this system has come into 
general use. 

THE MOLDING PROCESS 

If a gear blank made of some pliable material is forced into contact with a 
rack, as shown in Fig. 8, the rack tooth would conjugate teeth in the blank. 




FIG. 9. ACTION OF THE TOOL IN GENERATING A TOOTH. 



It does not matter what form is 
given the conjugating tooth, as long 
as it has a regular line of action ; all 
gears formed by it will interchange. 

The Bilgram spur and spiral gear 
generating machine operates upon 
this principle. See Fig. 9. The cut- 
ter A, which is a reciprocating or 
planing tool having the profile of a cor- 
rect rack tooth — namely, a truncated, 



Cutter 




FIG. 10. ACTION OF THE FELLOWS ' GEAR 
CUTTER. 



-Emery Wheel 




--• ^■•'■•■li ,- ' ^ J ^ 



Imaginary Rack 




Cutter 



FIG. II. GENERATION OF THE FELLOWS* 
GEAR-CUTTER TEETH. 



TOOTH PARTS 



9 



straight-sided wedge. While this tool reciprocates, it also travels slowly to 
the right, the blank meanwhile turning under it, the motion being that which 
would exist were the tool a rack tooth and the blank a gear. During this com- 
bined movement the tool cuts the tooth space in the manner indicated. In 
the Bilgram bevel-gear machine the tool does not move sidewise, the blank 
being rolled upon it as a complete gear might be rolled on a stationary 
rack, but in the spur-gear machine this action is reversed — the blank turning 
on a fixed center, while the tool moves over it, as it would be turned by a 
moving rack. 

The Fellows' gear shaper is designed on the same principle, but instead of 
a rack tooth as a planing tool, a gear of from 12 to 60 teeth is used, the motion 
of cutter and blank being the same as between gears in mesh. See Fig. 10. 
These cutters are ground to shape after being hardened as shown in Fig. 11, 
in which the emery wheel is shaped as the planing tool in Fig. 9. The cutter 
being ground taking the place of the gear. 



TO DRAW THE INVOLUTE CURVE 

The involute curve is constructed on the base circle as follows : Draw the 
pitch circle and through pitch point P, Fig. 12, draw the line of action at the 
required angle of obliquity. Tangent to this line draw the base circle. 

Divide the base circle into any num- 
ber of equal spaces, i', 2', 3', 4', 5', 6', 
as shown in Fig. 13. From each of 
these points draw lines intersecting at 





FIG. 12. LOCATING THE BASE CIRCLE. 



FIG. 13. DRAWING THE INVOLUTE. 



center 0. Draw lines I'-i, 2^-2, 3^-3, etc., tangent to base circle and at right 
angles with lines extending to center. Make the length of Hne I'-i equal to 
one of the divisions of base circle: Line 2^-2 equal to two divisions, line ^^-^ 
equal to three divisions, and so on. Then through points 1,2,3, 4? 5? 6, etc., 
trace the involute curve. Find a convenient radius, not necessarily on base 
circle, from which to draw the balance of the teeth, several radii sometimes 
being necessary to get the proper curve, especially for a small number of 
teeth. The involute curve does not extend below the base circle, for within 



lO 



AMERICAN MACHINIST GEAR BOOK 



the base circle it is simply a matter of obtaining sufficient clearance to 
avoid interference with the teeth of the mating gear. 




FIG. 14. DIAGRAM SHOWING HOW PROPER ACTION IS MAINTAINED AS THE GEAR AXES ARE 

SEPARATED. 




FIG. IS. GRAPHICAL DEMONSTRATION FOR INTERFERENCE OF SPUR GEARS. 

The involute curve is always the same for a given base circle diameter, but 
the angle of obliquity, or the path of tooth contact, is dependent upon the 



TOOTH PARTS 



II 



pitch diameter, so the effect of a change in center distance of engaging gears 
serves simply to alter the obUquity of the path of tooth contact. This is 
shown clearly in the diagram Fig. 14, in which the angles of obliquity for gear 
teeth of widely different pitch diameter, but the same base circle diameter, are 
depicted. It is this peculiarity of the involute system of gearing which 
permits a certain center adjustment of engaging gears of true involute form 
without destroying their correct tooth action. The greater the distance 
between pitch and base circles, the greater becomes the angle of obliquity. 

INTERFERENCE IN INVOLUTE GEARS 



The limitations and inaccuracies of the involute system are well explained 
in the following paragraphs by C. C. Stutz: 

While the general principles governing the interference of involute gears 
are well known, the following graphical 
demonstrations, formulas, and plotted 
diagrams may place this general infor- 
mation in more efl&cient form for the 
use of many. 

Fig. 16 shows a graphical demon- 
stration of the interference of a 5-pitch, 
15-tooth true involute form spur pinion 
and a 5-pitch, 48-tooth mating gear. 
The point F is the right-angled intersec- 
tion of a line drawn from the center of 
the pinion, and at an angle of 14^^ 
degrees with the common center line of 
the pinion and gear, with the line of 
pressure which is drawn through the 
point of tangency of the two pitch cir- 
cles and at an angle of 14^-^ degrees to 
the common tangent at that point. If 
this point falls within the addendum 
circle of the meshing gear, the tooth of 
the meshing gear will interfere from this 
point up to its addendum circle. There- 
fore the tooth from this point on 
the curve must be corrected to over- 
come it. 

If the point F falls on or outside of the addendum circle of the meshing 
gear no interference will result. The point F' for an angle of obliquity of 20 
degrees falls on the addendum circle and thus the gear and pinion indicated 
in the illustration would mesh without interference for this angle. 




FIG. 16. INTERFERENCE OF SPUR GEARS. 



12 AMERICAN MACHINIST GEAR BOOK 

FORMULA FOR LOCATING THE POINT OF INTERFERENCE OF SPUR GEARS 

Referring to Fig. i6: 

Let AF = c. AB = r^. AD = d. BE = ri. DE = f. 

a = the angle of tooth pressure. 

y = the distance from the center of the gear to the point at which in- 
terference begins. 

X = the distance from the point at which interference begins to the 
addendum circle of the gear measured along a radius. 

= the perpendicular distance from the point at which interference 
begins to the center line of the pinion and gear. 



Then 



Then 



Now 



ri = the pitch radius of the gear. 

^2 = the pitch radius of the pinion. 

D' = the pitch diameter of the gear. 

D = the outside diameter of the gear. 

c = r2 cos a, and 

d = c cos a = r2 cos^ a. 





f = ri -\- r2 — d. 




= ri -\- r2 {i — cos^ a), 


and 






= c sin a = r2 sin a cos a. 


Now 





y2 =p j^ Q2 and y = \/p +02 . 
Then by substituting 

y = V [^1 + ^2(1 = cos^ a )Y + (^2 sin a cos a)^ 
For a pressure angle of 143^^ degrees 

y = \/(^i + o.o627r2)2 + (0.2424^2)^, 
and 

D 

X = y. 

2 

For a pressure angle of 20 degrees 

y = 's/iji + 0.1169 ^2)^ + (0.3214 ^2)2, 

and 

D 

X = y. 

2 

Solving for x and y will give the point of interference for any particular 
case. 

DIAGRAM FOR LOCATION OF INTERFERENCE 

Fig. 17 shows a diagram giving the location of the point of the beginning of 
interference for one diametral pitch involute gears from 10 to 135 teeth mesh- 



TOOTH PARTS 



13 



ing with a 12-tooth pinion. The ordinates are the distances from the point 
where interference commences to the addendum circle of the gear measured 





















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Interference in Inches 



14 



AMERICAN MACHINIST GEAR BOOK 



along the radius. They correspond to the quantity x in the preceding equa- 
tions. From this point to the addendum circle the tooth outUne must be 
corrected. 

The upper curve A is for a pressure angle of 14}^ degrees and an adden- 
dum of 0.3183 X circular pitch. The second, B, is for the same pressure 
angle and a shorter addendum, 0.25 X circular pitch. 

This addendum factor is for what is known as the stubbed tooth standard, 
as proposed by the author on page 23. 

The third curve, C, is for a pressure angle of 20 degrees and an addendum of 
0.3183 X circular pitch, while the lowest one, D, is for the 20-degree angle and 
the stubbed tooth addendum. 

The diagram as plotted is for one diametral pitch. To find the corre- 
sponding ordinate for any other pitch divide the value given in the diagram by 
the required pitch. The quotient will be the distance desired. 

INTERFERENCE OF RACK AND PINION 

Interference will occur between the teeth of a rack and pinion when the 
point B, Fig. 18, which is the intersection of a perpendicular from the point O 
to the line of pressure AL falls inside of the rack addendum line EE. In the 




FIG. 18. INTERFERENCE OF GEAR AND RACK. 



figure the distance over which interference takes place is CD. It is usual 
practice to shorten the rack teeth by the amount of this interference and the 
following equations give an easy method of computing this distance. 



TOOTH PARTS 15 



Let N = the number of teeth in the pinion. 
p = the diametral pitch. 
r = the pitch radius. 
b = the radius of the base circle. 



Let 



a — the pressure angle. 

X = the distance necessary to shorten the addendum of the rack tooth 



and 



5 = the normal addendum of the rack tooth. 
Then 



s 


I 
-P' 






r 


p 






b 


= r cos a, 






OD 


= b cos a, 






OD 


= r cos- a, 






OC 


= r - s, 






X 


= OD - OC, 


and substituting 




= r cos^ a — 


(r 


-s) 




y^r 2 
= ^^(COS^ 

p 


a — 


p 


/V» 


I - >^iV(i 


: — 


cos^ a) 



Whence 

I - >^iV( 

P 
For a pressure angle of 143^^ degrees 

_ I - 0.03 13 57V 

X = -• 

P 
For a pressure angle of 20 degrees 

I — o.o584oiV 

X = • 

P 

Solving these equations we find that for the true involute form of tooth 
and a pressure angle of 14^ degrees interference between the teeth of rack and 
pinion begins with a pinion of 31 teeth. Similarly for a 20-degree pressure 
angle the interference begins with a pinion of 17 teeth. 

MODERN TOOTH FORM DEVELOPMENTS 

The introduction of generating machines, gear shapers and hobbers, for 
the rapid production of gears with teeth of the involute form has had a very 
marked effect upon gear development and has brought about radical innova- 
tions in the design and form of gear teeth. The limitations and inaccuracies 
developed in the practical application of the involute system necessitated 
considerable modification in the profile curvature of the teeth to avoid or to 



l6 AMERICAN MACHINIST GEAR BOOK 

overcome involute interference. Such modifications in the quantity produc- 
tion methods evolved to meet the demand for large numbers of high grade 
gears of specific sizes, such as is encountered in the automobile, machine tool 
and other fields, entailed no sacrifice in gear efficiency, as the peculiarity 
of true involute gearing which permits a certain center adjustment of gears 
without sacrifice of correct tooth action is no particular advantage. In such 
mechanisms, the gear centers are definitely fixed and the opportunity of 
center adjustment does not occur. 

The straight line of pressure peculiarity of involute gears, though an 
advantage in laying out (drawing) such gears, is also sacrificed in modern 
processes of gear production by generation — the profiles of the gear teeth 
being modified usually to avoid involute interference — so there remains no 
particular object in retaining the true involute form of tooth for a gear stand- 
ard. In fact, there have been developed other highly desirable tooth forms 
which possess distinctive advantages in the way of ease, economy and accu- 
racy in the actual commercial production of gearing. 

Before considering these later developments, however, it may be well to 
review the theory of gearing which led to the adoption of the involute system 
as a standard and the theory which justifies a radical departure from the 
involute form of gear tooth. The objective in all systems of toothed gearing 
is to secure as nearly as possible that positive uniform motion of the pitch 
surfaces of the gears upon which the efficiency of the mechanism is so largely 
dependent. This is basic and underlies the theory upon which the established 
tooth forms, such as the involute, were developed and also the modern tooth 
forms, such as that employed in the Williams system of internal gearing 
and the tooth form employed in the Anderson system of gear rolling. 

The chief difference in the older and more modern forms of gear teeth is 
not so much in the actual form of the teeth, although there is a very notice- 
able difference in form, as it is in the methods employed to attain the objective 
of positive uniform pitch surface motion. Before the advent of the gear 
generating machine and of the modern gear shaper, gear cutting entailed 
the machining of the individual teeth with the aid of formed cutters, the 
cutting of each tooth space presenting a separate machining process — a series 
of similar but nevertheless independent operations. In other words, an 
attempt was made to form the gear teeth so that they would transmit to 
the pitch surfaces of engaging gears a positive uniform motion. The gears 
being machined independently, this necessitated a standardization of gear 
tooth profiles in order that gears of various sizes might be run together 
satisfactorily. For various reasons, which need not be enlarged upon, the 
tooth form having profiles of involute curvature was found to possess dis- 
tinctive advantages and the involute system of gearing became the generally 
recognized standard. 

The development of gear generating machines brought about by the de- 
mand for more rapid and cheaper gear production contributed much to the 



TOOTH PARTS 1 7 

art of gear cutting and at the same time threw considerable light upon the 
requirements of tooth form needed for securing positive and uniform motion 
of pitch surfaces. These machines, whether they employ reciprocating 
cutters or hobs for machining the gear teeth, are so designed that the motion 
between the pitch surfaces of the cutting tool and the gear blank is positive 
and uniform during the machining operations and the teeth formed on the 
gear blanks are conjugately related. The result is that gears of various 
size engage one another in a manner to reproduce closely this positive uni- 
form motion of pitch surfaces. In short, the potent difference between gears 
with teeth cut individually by formed cutters and gears with generated teeth 
is that in the former case an attempt is made to proportion the teeth so that 
they will transmit to the gears in operation that desired positive uniform 
motion of pitch surfaces and in the latter case the positive uniform motion 
of pitch surfaces is employed in the formation of the teeth — in one case, 
development toward the objective and in the other, development backward 
from the objective. 

ADAPTATION OF GENERATION TO INVOLUTE GEARING 

The cutting of gear teeth of the involute form was naturally the first 
adaptation of gear generation, the cutting tooth section retaining the true 
involute rack form in the case of machines employing the principle of the 
Bilgram machine and subsequently in the case of the more modern gear 
hobbers and being of developed involute profile in the case of the Fellows' 
machine. With rack tooth machines and hobbers the generated teeth are 
obviously free from the involute interference of rack and pinion and that of 
externally meshing gears which is not so serious. In the case of pinion cutter 
machines the same result is secured as the pinion cutter teeth are really 
generated by an involute rack tool, or grinding wheel. See Fig. ii. The 
effect of this avoidance of tooth interference is that the profiles of the gener- 
ated teeth depart from the true involute curve over a greater or less distance, 
depending upon the number of gear teeth and their proportions — particu- 
larly in respect to the relative lengths of addendum and dedendum. 

In the case of internal gearing in which involute interference is more 
accentuated than in external gearing or in the rack and pinion, the modifica- 
tion of the profile curvature of the cutter teeth is of necessity considerably 
more pronounced. In fact, machining generation of internal gears is feasible 
only with pinion type cutters having cutting profiles departing quite notice- 
ably from the involute in curvature. The reason for this greater prevalence 
of involute interference in internal gearing is that the tooth profiles bounding 
the spaces between teeth in internal gears are of concave curvature, accentu- 
ating the interference occurring between rack and pinion just as the involute 
interference of externally meshing gears is relatively reduced by reason of 
the convex curvature of their tooth profiles. 

The concave profile curvature of internal gear teeth is well illustrated in 



i8 



AMERICAN MACHINIST GEAR BOOK 



Fig. 19, which depicts a method of interference correction for gears cut with 
formed tools, and is obvious as well in the equations derived to cover cases 
in which the pinion of an internal gear combination has less than 55 teeth. 
These equations estabhsh the length of the various radii for modifying the 
tooth profiles in order to correct for interference. 

INTERFERENCE OF INTERNAL GEAR AND PINION 




FIG. 19. INTERFERENCE OF INTERNAL GEAR AND PINION. 



Referring to Fig. 
Let 



19: 



Then 



ri = the pitch radius of gear. 
^2 = the pitch radius of pinion. 
b = the radius of the base circle. 
c = the radius of the correction circle. 
d = the radius of the rounding off circle. 
/ = the radius of the interference circle. 

= the radius of the tooth cutting. 
a = the pressure angle. 

b = ri cos a, 

1 = Yiih + n), 



C = 



r-i 



cos a 



= 



= Hru 



and 



' = V(;')'+(?)' 



These examples of involute interference and the modifications of involute 
tooth profiles in commercial production of gearing by generation are pre- 



TOOTH PARTS 



19 



sented to demonstrate the justification in departure from the former standard 
involute tooth form in the development of certain modern tooth forms which 
offer marked advantages in the way of reducing the cost of gear fabrication 
in large commercial production. Two of these newer forms of gear teeth 
are deserving of particular notice. 

WILLIAMS SYSTEM OF INTERNAL GEARING 

The gear teeth in the Williams development of internal gearing are of the 
involute rack tooth shape, the bounding profiles being straight lines. See Fig. 
20. The teeth of the engaging 
pinions have profiles of conju- 
gate curvature and are consider- 
ably more rugged and substantial 
than those of pinions for internal 
gears with teeth of in volute form. 
The effect of this modification is 
a curved line of pressure, or path 
of tooth contact, which materially lengthens the duration of contact be- 
tween pairs of pinion and gear teeth. 

ANDERSON ROLLED GEARS 

The latest attainment in gear producing methods, that of rolling gear 
teeth upon heated metal blanks by hardened die rolls under heavy pressure, 




Fig. 20. 




FtG.BT 



F/G.S8 



20 AMERICAN MACHINIST GEAR BOOK 

has also necessitated the adoption of a distinctive form for the functioning 
die roll teeth. As in the case of the Williams internal gear tooth, the simplest 
and most easily reproduced form— that of the involute rack tooth has been 
chosen. This form of tooth (Fig. 2 1) is in fact a " master form, " being basic 
for the involute system of gearing as well as for those of the Williams internal 
gears and Anderson rolled gears. 

In the production of gears by the hot rolling method, the ''master form" 
is employed for the teeth of the die roll used in forging the teeth on the heated 
gear blanks and is not, consequently, the exact section of the teeth formed by 
the moulding generation process. The latter are developed forms which vary 
in profile curvature somewhat with the class of die roll employed, but they 
are all developments from the one basic " master form." In the cases of spur, 
hehcal and herringbone gears with their pitch surfaces concentric with their 
axes, the tooth form is very similar to the involute, being modified only by a 
slight increase in curvature of both addendum and dedendum sections of the 
tooth profiles. The modification is not apparent to the eye, as in the actual 
fabrication of gears by this method the number of die roll teeth is consid- 
erably greater than the number of teeth rolled on the gear blank, so that the 
modification in the rolled tooth profile is negligible for all practical purposes. 

In the rolling of bevel gears of any type, the same negligible modification 
occurs when the die roll is of the flat bevel gear form, but no modification 
from the octoid form of tooth takes place when the die roll is of the crown 
gear form— see Figs. 21-28 for diagrammatic depictions of Anderson die roll 
teeth. 

EXISTING TOOTH STANDARDS— BROWN & SHARPE'S 

The Brown & Sharpe system is perhaps the best known; the angle of 
obliquity being 143.^ degrees. 



Addendum, = 0.3183^^ or 



I 
P 



Dedendum, = 0.3683/?^ or ^'^^' 

P 



Working depth, = 0.6366/?^ or 



2 
P 

Whole depth, = o.6866/>i or ^^^ 

P 

Clearance, = o.o$p^ or 

P 

In which p^ = circular pitch, and p = diametral pitch. 

GRANT'S 

The Grant system has an angle of obliquity of 15 degrees, otherwise it is 
the same as Brown & Sharpe's. This system is used on the Bilgram generator. 



TOOTH PARTS 21 

SELLERS* 

Wm. Sellers & Co. adopted a form of tooth some 32 years ago in which 
the angle of obliquity was 20 degrees, with an addendum of 0.3 and a clear- 
ance of 0.05 of the circular pitch. 

HUNT'S 

The C. W. Hunt Co. have a standard in which the angle of obliquity is 
143^^ degrees; the tooth parts being as follows: 



Addendum, = 0.25/?^, or 

Dedendum, = 0.30^^ or 

Working depth, = o.^op^, or 

Whole depth, = o.55/>S or 



0-7854 

P 
0.9424 

P 
1-5708 

P 
1. 71278 



Clearance, = o.oKp^, or ' 

P 

THE AUTHOR'S 

This system, presented in connection with a discussion of an interchange- 
able involute gear-tooth system at the December, 1908, meeting of the 
American Society of Mechanical Engineers, was originally published in 
American Machinist, June 6, 1907. Angle of obliquity is 20 degrees. 
Balance of the tooth parts being the same as the Hunt system described above. 

FELLOWS' 

The stubbed tooth adopted by the Fellows Gear Shaper Company has an 
angle of obliquity of 20 degrees. The tooth parts, however, do not bear a 
definite relation to the pitch; the addendum being made to correspond to a 
diametral pitch one or two sizes finer, as: 

Actual pitch _ 2 23^^ 3 4 5 7 8 10 12 14 
Pitch depth 2^^ 3 4 5 7 9 10 12 14 18 

The upper figures indicate the diametral pitch for tooth spacing and the 
lower figures indicate the diametral pitch from which the depth is taken. 

In this system the addendum varies from 0.264 to 0.226 of the circular 
pitch; 0.25, which is the addendum for the Hunt and the author's standard, 
is a rough mean. 

The author's standard tooth is shown in Fig. 30 for comparison with the 
i4j^-degree standard in Fig. 29. 

Wilfred Lewis discussed tooth standards before the American Society of 
Mechanical Engineers, 1900, as follows: 



22 



AMERICAN MACHINIST GEAR BOOK 



"About 30 years ago, when I first began to study the subject, the only 
system of gearing that stood in much favor with machine-tool builders was 
the cycloidal. 




Fig. 30 

Proposed Stubbed Involute Tooth Shape 

12 Teeth, 20 Degrees. 

Addendum =0.25 X Circular Pitch- 

COMPARATIVE FORMS OF 141^-DEGREE AND 20-DEGREE STANDARDS. 

"For some time thereafter William Sellers & Co., with whom I was con- 
nected, continued to use cylindrical gearing made by cutters of the true cycloi- 
dal shape, but the well-known objection to this form of tooth began to be 
felt, and possibly 25 years ago my attention was turned to the advantages of 
an involute system. The involute systems in use at that time were the ones 
here described as standard, having 14^^^ degrees' obliquity, and another 
recommended by Willis having an obliquity of 15 degrees. Neither of these 
satisfied the requirements of an interchangeable system, and with some 
hesitation I recommended a 20-degree system, which was adopted by William 
Sellers & Co., and has worked to their satisfaction ever since. I did not at 
that time have quite the courage of my convictions that the obliquity should 
be 223^^ degrees or one-fourth of a right angle. Possibly, however, the obli- 
quity of 20 degrees may still be justified by reducing the addendum from a 
value of one to some fraction thereof, but I would not undertake at this time 
to say which of the two methods I would prefer. 

"I brought up the same question nine years ago before the Engineers' Club 
of Philadelphia, and said at that time that a committee ought to be appointed 
to investigate and report on an interchangeable system of gearing. We have 
an interchangeable system of screw threads, of which everybody knows the 
advantage, and there is no reason why we should not have a standard system 
of gearing, so that any gear of a given pitch will run with any other gear of 
the same pitch." 



TOOTH PARTS 



23 



A UNIVERSAL STANDARD 

At that time, it was felt that to do away with the then existing multiplicity 
of standards, especially in connection with the involute form of tooth, and 
adopt an acceptable standard gear-tooth system would be a much-needed 
reform. A special committee was appointed by the American Society of 
Mechanical Engineers, under the chairmanship of Wilfred Lewis, to bring 
this about, but the committee accomplished little and finally gave up the work. 
More recently other committees and other organizations have attempted the 
same reform, but to little avail. The modifications of tooth section found 
necessary in automobile work and the development of new and efficient forms 
of gear teeth has had much to do with this and with the attainment of a 
practical and highly efficient process of rolling gear teeth on heated blanks — • 
the Anderson process, which will be discussed in a subsequent section — the 
acceptable standard today will of necessity have to disregard the form of the 
tooth and permit considerable latitude in overall tooth dimensions. 

It is desirable, however, to have a gear- tooth system which will permit 
some uniformity in formulas for computations, even though profile curvatures, 
etc., may differ. Such an elastic system is even now generally recognized and 
the subsequent equations and calculations will be based on present day prac- 
tice as developed for the involute system. The illustrations will also depict 
involute gearing for the most part, the new gear systems being discussed 
specifically and individually. 

MODIFIED TEETH 

A common method of modifying the involute tooth to avoid either inter- 
ference, undercutting, or the necessity of departing from the true outline is by 
shortening the dedendum and lengthening 
the addendum of the pinion tooth. The 
opposite treatment is given the gear tooth, 
the dedendum being made deeper to ac- 
commodate the added addendum of the 
pinion and the addendum of the gear cor- 
respondingly short. This method is 
employed on all bevel-gear generating 
machines for angles less than 20 degrees 
to avoid interference, the amount of cor- 
rection depending, of course, upon both the number of teeth being cut and 
the number of teeth in the engaging gear, or, in other words, depending upon 
the position of the base line. 

On bevel-gear generating machines it is the practice to make no modifica- 
tion in the angle for a 20-degree tooth when cutting a depth equal to 0.6866^'. 
For this depth of tooth and a pressure angle of 20 degrees interference begin- 
ning at 17 teeth, enough roll, however, can be given the blank to allow the 




12 Teeth 

3 Diarr(etral Pitch 

2V2 Inch Bore 



FIG. 31. SHORTENING THE DEDENDUM 
TO STRENGTHEN KEY WAY. 



24 



AMERICAN MACHINIST GEAR BOOK 



generating tool to undercut the flank of the tooth, and avoid interference with- 
out any correction of the tooth parts. This is not the case, however, when 
cutting the standard 143^^ or 15 degrees on account of limitation in the move- 
ments of the machine. This modification in the tooth parts for bevel gears is 
accomplished by shifting the face angles and outside diameters, the pinion 
being enlarged and the gear reduced. 

The dedendum of the pinion is sometimes shortened for another reason: 
Often the bore is so large as to leave insufficient stock between the bottom of 
the teeth and the keyseat. See Fig. 3 1 . When the pinion cannot be enlarged 
or the bore reduced the only possible recourse is to shorten the dedendum, 
taking the amount shortened from the point of the gear tooth. This practice 
is not to be recommended although extensively used; it would be much better 
to apply the short tooth of increased obliquity to such cases. 

THE OCTOID 

All bevel-gear generating machines operate on the octoid system, and not 
the involute, as is generally supposed. 





I 



Involute Tooth 




Octoid Tooth 
Fig. 33. 



American JIfachinist 



FIG. 32. GENERATING THE OCTOID TOOTH. 

An involute spur gear may be generated by the action of a tool represent- 
ing the rack tooth, as illustrated by Fig. 9. In generating a bevel gear, how- 
ever, the tool representing the engaging rack tooth must always travel toward 
the apex of the gear being cut, swinging in a partial circle instead of travelling 
on a straight line in the direction of the rotation of the gear, as is the case 
when cutting a spur gear. The base of the bevel-gear tooth is, therefore, a 
crown gear instead of a rack. 

An involute crown gear theoretically correct will have curved instead of 
straight sides as shown in Fig. 32. As it is not practical to make the gener- 
ating tools this peculiar shape, they are made straight sided and the octoid 
tooth is the result. 



TOOTH PARTS 



25 



THE LINE OF ACTION 

There is a definite relation between the circle or line which will describe 
the tooth outline and the line of action. Thus, if the line of action is in the 
form of a circle, as shown in Fig. 34, that circle of which this line is a segment 
will describe the tooth outline if rolled upon the pitch circle. The difficulties 
encountered in the general application of this law are well illustrated by 
George B. Grant in section 32 of his ''Treatise on Gear Wheels," as follows: 
*'This accidental and occasionally useful feature of the rolled curve has 
generally been made to serve as a basis for the general theory of the tooth 
curve, and it is responsible for the usually clumsy and limited treatment of 
that theory. The general law is simple 
enough to define, but it is so difficult to 
apply, that but one tooth curve, the 
cycloidal, which happens to have the 
circle for a roller, can be intelligently 
handled by it, and the natural result is 
that that curve has received the bulk of 
the attention. 

For example, the simplest and best 
of all the odontoids (pure form of tooth 
curve), the involute, is entirely beyond 
its reach, because its roller is the loga- 
rithmic spiral, a transcendental curve 
that can be reached only by the higher mathematics. 

No tooth curve, which, like the involute, crosses the pitch line at any 
angle but a right angle, can be traced by a point in a simple curve. The 
tracing point must be the pole of a spiral, and therefore a mechanical 
impossibility. A practicable rolled odontoid must cross the pitch line at 
right angles. 

To use the rolled curve theory as a base of operations will confine the dis- 
cussion to the cycloidal tooth, for the involute can only be reached by aban- 
doning its true logarithmic roller, and taking advantage of one of its peculiar 
properties, and the segmental, sinusoidal, parabolic, and pin tooth, as well as 
most other available odontoids, cannot be discussed at all." 




FIG. 34. RELATION OF THE LINE OF 
ACTION TO THE DESCRIBING CIRCLE. 



THE LAW OF TOOTH CONTACT 

To transmit uniform motion, any form of tooth curve is subject to this 
general law: "The common normal to the tooth must pass through the pitch 
point." That is, a line drawn from the pitch point P through the contact 
point of any pair of teeth, as at h, must be at right angles or normal to the 
engaging portions of both teeth. See Fig. 35. 

In the involute system the line of action always passes through the pitch 
point P, and the engaging teeth take their base from the points / and y, 



26 



AMERICAN MACHINIST GEAR BOOK 



where the Hne of action intersects the base circles. Conversely, a line drawn 
from the instantaneous radii of any two teeth engaged will pass through their 
point of contact if the teeth are correctly formed. For example: In Fig. 35, 
the point of contact between the teeth C and Z) is at 6, on the line of action, 




FIG. 35. THE ARC OF ACTION. 

the radius of the engaged portion of the tooth C is at /, and the radius of the 
tooth D is at y, fulfilling the required conditions. 

THE ARC OF ACTION 

The tooth action between two gears is between the points a and 6, at which 
points the line of action intersects the addendum circles of the two gears. 
The actual length of contact is along the pitch lines occupied by the teeth 
whose addendum circles intersect the line of action, or between the points 
c and d. See Fig. 35. 

The distance P — d passed over while the point of contact approaches the 
pitch point is the arc of approach, the distance P — c passed over as the point 
of contact leaves the pitch point is the arc of recess. 

By increasing the addendum of the driving gear the arc of approach is 
reduced and the arc of recess is increased. The friction of the arc of approach 
is greater than in the arc of recess. 

THE BUTTRESSED TOOTH 

The buttressed tooth shown in Fig. 36 is described by Professor Robert 
Willis in a paper published in the Transactions of the Institute of Civil Engi- 
neers, London, 1838. It is apparent that the object is to obtain a strong tooth 



TOOTH PARTS 



27 



for a pair of gears operating continuously in one direction. This is accom- 
plished by increasing the angle of obliquity of the back of the tooth, the face 
of the tooth being any angle desired. If the back of "the tooth is correctly 




11 Teeth 
I 2J^ Inch Pitch 



FIG. 36. THE BUTTRESSED TOOTH. 

formed the gears will operate satisfactorily in either direction although with 
an increased pressure on their bearings when using the back face of the teeth 
owing to the increased obliquity of action. For many purposes there is no 




FIG. 37. BUTTRESSED TEETH IN CONTACT. 



objection to this, and it is a great wonder that this tooth is not more exten- 
sively used. 

Of course, there must be a limit to the angle of the back of the tooth. For 
practical purposes the curve at the top of tooth at the back should not extend 

farther than the center line of the tooth; for an addendum of - or o.6866/)i, 



28 



AMERICAN MACHINIST GEAR BOOK 



this will occur at an angle of about 32 degrees, 
subject the tooth to .breakage at the point, 
buttressed tooth gears in contact. 



A greater angle than this will 
In Fig. 37 is shown a pair of 



STEPPED GEARS 

A stepped spur gear consists of two or more gears keyed to the same shaft, 
the teeth on each gear being slightly advanced. If mated with a simillar gear 
the tooth contact will be increased, which increases the smoothness of action. 
A common form of this type of gear is that of two gears cast in one piece with 
a separating shroud. For a cut gear there must be a groove turned between 
the faces of sufficient width to allow the planing tool or cutter to clear. A 
tooth is placed opposite a space, when the gear is made in two sections. 

HUNTING TOOTH 

It has been customary to make a pair of cast tooth gears with a hunting 
tooth, in order that each tooth would engage all of the teeth in the mating 
gear, the idea being that they would eventually be worn into some indefinite 
but true shape. Some designers have even gone so far as to specify a pair of 
*' hunting- tooth miter gears." That is, one ''miter" gear would have, say, 
24 teeth and its mate 25 teeth. 

There never was any call for the introduction of the hunting tooth even in 
cast gears, but in properly cut gears any excuse for its use has certainly ceased 
to exist. 

TEMPLET MAKING 

In making the templets for gear teeth there are several points of impor- 
tance to be kept in mind, namely: 

Templets should be made of light sheet 
steel instead of zinc which is often employed; 
the surface of steel should be coppered by 
the application of blue vitriol. 

For spiral or worm gears, templets should 
always be made for the normal pitch. 

For spacing and tooth thickness, always 
use chordal measurements. Check the 
chordal distance over the end teeth of tem- 
plet. This is of the utmost importance. 

Put enough teeth in the templet to show 
the entire tooth action, and try the tem- 
plets on centers before making up the cut- 
ters or formers. 

It is a good idea to make a standard templet of each pitch as they are 
required, to try out other templets that must be made later on. 




FIG. 38. TOOTH OUTLINE AS PHOTO- 
GRAPHED FROM LARGE SCALE 
DRAWING. 



TOOTH PARTS 



29 



When a templet is required for a fine pitch gear it is good practice to lay 
out the teeth on white paper 10 or even 20 times the actual size and reduce by 
photography. On this drawing the center should be plainly marked and 
inclosed in a heavy circle, also a short section of the pitch line should be made 
heavy with a connecting radial line indicating the radius of pitch circle. 

If the pitch radius required is ij^ inches, it should be made, say, 15 inches 
on the drawing. The drawing is then photographed, the camera being set 
until the radial line, which was drawn 15 inches, measures i)-^ inches on the 
ground glass. See Fig. 38. 

DEFINITION OF PITCHES 

Diametral pitch is the number of teeth to each inch of the pitch diameter. 
Circular pitch is the distance from the edge of one tooth to the corre- 
sponding edge of the next tooth measured along the pitch circle. 



Addendum 




FIG. 39. TOOTH PARTS. 



30 



AMERICAN MACHINIST GEAR BOOK 



DIAMETRAL 
PITCH 


CIRCULAR 
PITCH 


THICKNESS OF 

TOOTH OF 

PITCH LINE 


WHOLE DEPTH 


DEDENDUM 


ADDENDUM 


^ 


6.2832" 


3.1416" 


4.3142" 


2.3142" 


2.0000" 


% 


4.1888 


2.0944 


2.8761 


1.5728 


T-Zm 


I 


3.1416 


1.5708 


2.1571 


I.1571 


1 .0000 


iM 


2.5133 


1.2566 


1-7257 


0.9257 


0.8000 


^¥2 


2.0944 


1.0472 


I.4381 


0.7714 


0.6666 


iH 


1-7952 


0.8976 


1.2326 


0.6612 


0.5714 


2 


1.5708 


0.7854 


1.0785 


0.5785 


0.5000 


2H 


1-3963 


0.6981 


0.9587 


0.5143 


0.4444 


2y2 


1.2566 


0.6283 


0.8628 


0.4628 


0.4000 


2% 


1. 1424 


0.5712 


0.7844 


0.4208 


0.3636 


3_ 


1.0472 


0.5236 


0.7190 


0.3857 


o.2,2>3^ 


3>^ 


0.8976 


0.4488 


0.6163 


0.3306 


0.2857 


4 


0.7854 


0.3927 


0.5393 


0.2893 


0.2500 


5 


0.6283 


0.3142 


0.4314 


0.2314 


0.2000 


6 


0.5236 


0.2618 


0.3595 


0.1928 


0.1666 


7 


0.4488 


0.2244 


0.3081 


0.1653 


0.1429 


S 


e.3927 


0.1963 


0.2696 


0.1446 


0.1250 


9 


0.3491 


0.1745 


0.2397 


0.1286 


O.IIII 


10 


0.3142 


O.1571 


0.2157 


O.II57 


O.IOOO 


II 


0.2856 


0.1428 


O.I961 


0.1052 


0.0909 


12 


0.2618 


0.1309 


0.1798 


0.0964 


0.0833 


13 


0.2417 


0.1208 


0.1659 


0.0890 


0.0769 


14 


0.2244 


O.II22 


O.1541 


0.0826 


0.0714 


IS 


0.2094 


0.1047 


0.1438 


0.0771 


0.0666 


16 


0.1963 


0.0982 


0.1348 


0.0723 


0.0625 


17 


0.1848 


0.0924 


0.1269 


0.0681 


0.0588 


18 


0.1 745 


0.0873 


O.II98 


0.0643 


0.055s 


19 


0.1653 


0.0827 


O.II35 


0.0609 


0.0526 


20 


0.1571 


0.0785 


0.1079 


0.0579 


0.0500 


22 


0.1428 


0.0714 


0.0980 


0.0526 


0.0455 


24 


0.1309 


0.0654 


0.0898 


0.0482 


0.0417 


26 


0.1208 


0.0604 


0.0829 


0.0445 


0.0385 


28 


0.1122 


0.0561 


0.0770 


0.0413 


0.0357 


30 


0.1047 


0.0524 


0.0719 


0.0386 


0.0333 


32 


0.0982 


0.0491 


0.0674 


0.0362 


0.0312 


34 


0.0924 


0.0462 


0.0634 


0.0340 


0.0294 


36 


0.0873 


0.0436 


0.0599 


0.0321 


0.0278 


38 


0.0827 


0.0413 


0.0568 


0.0304 


0.0263 


40 


0.0785 


0.0393 


0.0539 


0.0289 


0.0250 


42 


0.0748 


0.0374 


0.0514 


0.0275 


0.0238 


44 


0.0714 


0.0357 


0.0490 


0.0263 


0.0227 


46 


0.0683 


0.0341 


0.0469 


0.0252 


0.0217 


48 


0.0654 


0.0327 


0.0449 


0.0241 


0.0208 


SO 


0.0628 


0.0314 


0.0431 


0.0231 


0.0200 


S6 


0.0561 


0.0280 


0.9385 


0.0207 


0.0178 


60 


0.0524 


0.0262 


0.0360 


0.0193 


0.0166 



Table i — Diametral Pitch 
Relation between Diametral Pitch and Circular Pitchy with corresponding Tooth Dimensions 



TOOTH PARTS 



31 



CIRCULAR 
PITCH 


DIAMETRAL 
PITCH 


THICKNESS OF 

TOOTH OF 

PITCH LINE 


WHOLE DEPTH 


DEDENDUM 


ADDENDUM 


6 " 


0.5236 


3.0000" 


4.II96" 


2.2098" 


1.9098" 


5 " 


0.6283 


2.5000 


3-4330 


I.8415 


I-5915 


4 " 


0.7854 


2.0000 


2.7464 


1.4732 


1.2732 


3^;; 


0.8976 


1.7500 


2.4031 


1.2890 


I.II40 


3 " 


1.0472 


1.5000 


2.0598 


1.1049 


0.9550 


2%" 


1. 1424 


1-3750 


1.8882 


1.0028 


0.8754 


2^i" 


1.2566 


1.2500 


1-7165 


0.9207 


0.7958 


2^" 


1-3963 


1. 1250 


1-5449 


0.8287 


0.7162 


2 " 


1.5708 


1 .0000 


1-3732 


0.7366 


0.6366 


i3^" 


1-6755 


0-9375 


1.2874 


0.6906 


0.5968 


i^" 


1-7952 


0.8750 


1. 2016 


0.6445 


0.5570 


i^" 


1-9333 


0.8125 


1.1158 


0.5985 


0.5173 


i>^" 


2.0944 


0.7500 


1.0299 


0.5525 


0.4775 


lYe" 


2.1855 


0.7187 


0.9870 


0.5294 


0.4576 


i^" 


2.2848 


0.6875 . 


0.9441 


0.5064 


0.4377 


i^e" 


2.3936 


0.6562 


0.9012 


0.4837 


0.4178 


iM" 


2-5133 


0.6250 


0.8583 


0.4604 


0.3979 


1^6" 


2.6465 


0.5937 


0.8156 


0.4374 


0.3780 


1,^" 


2.7925 


0.5625 


0.7724 


0.4143 


0.3581 


i^Te" 


2.9568 


0.5312 


0.7295 


0.3913 


0.3382 


I " 


3-1416 


0.5000 


0.6866 


0.3683 


0.3183 


%" 


3-3510 


0.4687 


0.6437 


0.3453 


0.2984 


Vs" 


3-5904 


0.4375 


0.6007 


0.3223 


0.2785 


%" 


3.8666 


0.4062 


0.5579 


0.2993 


0.2586 


%" 


4.1888 


0.3750 


0.5150 


0.2762 


0.2387 


%" 


4.5696 


0.3437 


0.4720 


0.2532 


0.2189 


%" 


5.0265 


0.3125 


0.4291 


0.2301 


0.1989 


%" 


5-5851 


0.2812 


0.3862 


0.2071 


0.1790 


y^' 


6.2832 


0.2500 


0.3433 


0.1842 


0.1592 


JTe" 


7.1808 


0.2187 


0.3003 


O.161I 


0.1393 


I" 


7.8540 


0.2000 


0.2746 


0.1473 


0.1273 


%" 


8.3776 


0.1875 


0.2575 


O.I381 


O.II94 


Vs" 


9.4248 


0.1666 


0.2287 


0.1228 


O.I061 


Xe" 


10.0531 


0.1562 


0.2146 


O.II51 


0.0995 


V 


10.9956 


0.1429 


0.1962 


0.1052 


0.0909 


H" 


12.5664 


0.1250 


0.1716 


0.0921 


0.0796 


r' 


14.1372 


O.IIII 


0.1526 


0.0818 


0.0707 


*" 


15.7080 


O.IOOO 


0.1373 


0.0737 


0.0637 


^e" 


16.7552 


0.0937 


0.1287 


0.0690 


0.0592 


1/" 


18.8496 


0.0833 


0.1144 


0.0614 


0.0531 


1" 


21.9911 


0.0714 


0.0981 


0.0526 


0.0455 


3^" 


25.1327 


0.0625 


0.0858 


0.0460 


0.0398 


i" 


28.2743 


0.0555 


0.0763 


0.0409 


0.0354 


h" 


31-4159 


0.0500 


0.0687 


0.0368 


0.0318 


X" 


50.265s 


0.0312 


0.0429 


0.0230 


0.0199 



Table 2 — Circular Pitch 
Relation between Circular Pitch and Diametral Pitch, with corresponding Tooth Dimensions 



32 



AMERICAN MACHINIST GEAR BOOK 



AAA 

20 D. P. 
0.1571 Inch C. P. 



18 D. P. 
0.1745 Inch C. P. 



pJXfl 

16 D. P. 
0.1963 Inch C. P. 



AAA 

14 D. P. 
0.2244 Inch C. P. 



12 D. P. 

0.2618 Inch C. P. 




10 D. P. 
0.3142 Inch C. P. 




9 D. P. 
0.3491 Inch C. P. 




8 D. P. 
0.3927 Inch C. P. 




7 D. P. 
0.4488 Inch C. P. 





6 D.P. 
0.5236 Inch C. P. 



5 D.P. 
0.6283 Inch C. P. 




4 D.P. 
0.7854 Inch C. P. 




3D. P. 

1.0472 Inch C. P. 



COMPARATIVE SIZES OF GEAR TEETH — INVOLUTE FORM. 



TOOTH PARTS 



33 




2^2 D. P. 
1.2566 In. C.P. 




2 D.P. 
1.5708 In. C.P. 




V4 D.P. 
1.7952 In. C.P. 




m D.P. 

2.0944 In. C.P. 

COMPARATIVE SIZES OF GEAR TEETH — INVOLUTE FORMS. 



34 



AMERICAN MACHINIST GEAR BOOK 




D.P. 
2.5133 Inch C. P. 




1 D.P. 
3.1416 Inch C. P. 



COMPARATIVE SIZES OF GEAR TEETH — INVOLUTE FORM. 



TOOTH PARTS 



35 



NO. 


PITCH 


NO. 


PITCH 


NO. 


PITCH 


NO. 


PITCH 


TEETH 


DIAMETER 


TEETH 


DIAMETER 


TEETH 


DIAMETER 


TEETH 


DIAMETER 


8 


2.550 


43 


13.687 


78 


24.828 


"3 


35-968 


9 


2.870 


44 


14.006 


79 


25.146 


114 


36.286 


lO 


3-183 


45 


14-324 


80 


25-465 


115 


36.605 


II 


3-50I 


46 


14.642 


81 


25-783 


116 


36.923 


12 


3.820 


47 


14.961 


82 


26.101 


117 


37.241 


13 


4.138 


48 


15-279 


&3 


26.420 


118 


37-560 


14 


4456 


49 


15-597 


84 


26.738 


119 


37-878 


15 


4-775 


50 


15-915 


85 


27.056 


120 


38.196 


i6 


5-093 


51 


16.234 


86 


27-375 


121 


38.514 


17 


5-4II 


52 


16.552 


S7 


27.693 


122 


38.833 


i8 


5-730 


53 


1 6. -8 70 


88 


28.011 


123 


39-151 


19 


6.048 


54 


17.189 


89 


28.330 


124 


39-469 


20 


6.366 


55 


17-507 


90 


28.648 


125 


39-788 


21 


6.684 


56 


17.825 


91 


28.966 


126 


40.106 


22 


7-003 


57 


18.144 


92 


29.284 


127 


40.424 


23 


7.321 


58 


18.462 


93 


29.603 


128 


40.743 


24 


7-639 


59 


18.780 


94 


29.921 


129 


41.061 


25 


7-958 


60 


19.099 


95 


30.239 


130 


41-379 


26 


8.276 


61 


19.417 


96 


30.558 


131 


41.697 


27 


8.594 


62 


19-735 


97 


30.876 


132 


42.016 


28 


8.913 


63 


20.053 


98 


31.194 


133 


42.334 


29 


9.231 


64 


20.372 


99 


31-513 


134 


42.652 


30 


9-549 


65 


20.690 


100 


31.831 


135 


42.971 


31 


9.868 


66 


21.008 


lOI 


32.148 


136 


43.289 


32 


10.186 


67 


21.327 


102 


32.468 


137 


43.607 


33 


10.504 


68 


21.645 


103 


32.785 


138 


43.926 


34 


10.822 


69 


21.963 


104 


33-103 


139 


44.243 


35 


II. 141 


70 


22.282 


105 


33-421 


140 


44.562 


36 


11-459 


71 


22.600 


106 


33.740 


141 


44.881 


37 


11.777 


72 


22.918 


107 


34.058 


142 


45.199 


38 


12.096 


73 


23.237 


108 


34-376 


143 


45-517 


39 


12.414 


74 


23-555 


109 


34-695 


144 


45-835 


40 


12.732 


75 


23-873 


no 


35-013 


145 


46.154 


41 


13-051 


76 


24.192 


III 


35-331 


146 


46.472 


42 


13.369 


77 


24.510 


112 


35-650 


147 


46.790 



Table 3 — Pitch Diameters for One-Inch Circular Pitch 

Teeth from 8 to 147 

FOR any other pitch MULTIPLY BY THAT PITCH 



METRIC PITCH 



The module is the addendum, or the pitch diameter in millimeters divided 
by the number of teeth in the gear. 



S6 



AMERICAN MACHINIST GEAR BOOK 



The pitch diameter in millimeters is the module multiplied by the number 
of teeth in the gear. All calculations are in millimeters. 

M = module (addendum) 
D' = pitch diameter 
D = outside diameter 
N = number of teeth 
W = working depth of tooth 
W = whole depth of tooth 

/ = thickness of tooth at pitch line 

/ = clearness 

r = root 



^ = 1^ or ITT-I 

N N -i- 2 

AT ^' D 

^ = T7 or ii> ~~ 2 

M M 

t = M 1.5708, 



W = 2M 

M 1.5708 



/ = 



10 



D = (iV + 2) M 
W' = W +J 
} or M 0.157, r = M -\- f 





ENGLISH 




ENGLISH 




ENGLISH 


MODULE 


DIAMETRAL 


MODULE 


DIAMETRAL 


MODULE 


DIAMETRAL 




PITCH 




PITCH 




PITCH 


o-S 


50.800 






7 


3.628 


I 


25.400 


3 


8.466 


8 


3.175 


1.25 


20.320 


Z-S 


7.257 


9 


2.822 


i-S 


16.933 


4 


6.350 


10 


2.540 


1-75 


. 14-514 


4.5 


5.644 


II 


2.309 


2 


12.700 


S 


5.080 


12 


2. 117 


2.25 


11.288 


5-5 


4.618 


14 


I.814 


2.5 


10.160 


6 


4.233 


16 


1.587 


2.75 


9.236 











Module in Millimeters 
Table 4 — Pitches Commonly Used 



CHORDAL PITCH 

The chordal pitch is the shortest distance between two teeth measured on 
the pitch line, in other words, the distance to which the dividers would be set 
to space around the gear on the pitch line. This pitch is not used except for 
laying out large gears and segments that cannot be cut on a gear cutter. For 
such cases, also for laying out templets, it is absolutely necessary to use the 
chordal pitch, as the chordal pitch of the pinion is different from the chordal 
pitch of the gear, the circular pitch of each being equal. 

A^ = number of teeth, 
C = chordal pitch. 
D' = pitch diameter, 
e = sine of one half of angle subtended by side at center. 



TOOTH PARTS 



37 



e = sine 



i8o^ 

N 



D' = 



C 



C = D' e, or D' sine 



1 80^ 

N 



Table 5, diameters for chordal pitch, will be found useful for sprocket 
gears. 



NO. 


PITCH 


NO. 


PITCH 


NO. 


PITCH 


NO. 


PITCH 


TEETH 


DIAMETER 


TEETH 


DIAMETER 


TEETH 


DIAMETER 


TEETH 


DIAMETER 


4 


1-414 


39 


12.427 


74 


23.562 


109 


34-701 


5 


1. 701 


40 


12.746 


75 


23.880 


no 


35-019 


6 


2.000 


41 


13.064 


76 


24.198 


III 


35-337 


7 


2.305 


42 


13-382 


17 


24.517 


112 


35-655 


8 


2.613 


43 


13.699 


78 


24.835 


113 


35-974 


9 


2.924 


44 


14.018 


79 


25-153 


114 


36.292 


10 


2>-^2,(> 


45 


14-335 


80 


25-471 


115 


36.610 


II 


3-549 


46 


14-653 


81 


25.790 


116 


36.929 


12 


3.864 


47 


14.972 


82 


26.108 


117 


37-247 


13 


4.179 


48 


15.291 


83 


26.426 


118 


37-565 


14 


4.494 


49 


15.608 


84 


26.744 


119 


37-883 


15 


4.810 


50 


15-927 


85 


27.063 


120 


38.202 


16 


5.126 


51 


16.244 


86 


27-381 


121 


38.520 


17 


5-442 


52 


16.562 


87 


27.699 


122 


38.838 


18 


5-759 


53 


16.880 


88 


28.017 


123 


39-156 


19 


6.076 


54 


17.200 


89 


28.335 


124 


39.475 


20 


6.392 


55 


17.516 


90 


28.654 


125 


39.793 


21 


6.710 


56 


17-835 


91 


28.972 


126 


40.1 II 


22 


7.027 


57 


18.152 


92 


29.290 


127 


40.429 


23 


7-344 


58 


18.471 


93 


29.608 


128 


40.748 


24 


7.661 


59 


18.789 


94 


29.927 


129 


41.066 


25 


7-979 


60 


19.107 


95 


30-245 


130 


41.384 


26 


8.297 


61 


19.425 


96 


30-563 


131 


41-703 


27 


8.614 


62 


19.744 


97 


30.881 


132 


42.021 


28 


8.931 


(>2, 


20.062 


98 


31.200 


133 


42.339 


29 


9.249 


64 


20.380 


99 


31.518 


134 


42.657 


30 


9-567 


65 


20.698 


100 


31-836 


135 


42.976 


31 


9.884 


66 


21.016 


lOI 


32-154 


136 


43-294 


32 


10.202 


67 


21-335 


102 


32-473 


137 


43.612 


33 


10.520 


68 


21.653 


103 


32.791 


138 


43.931 


34 


10.838 


69 


21.971 


104 


33-109 


139 


44.249 


35 


II. 156 


70 


22.289 


105 


33-428 


140 


44.567 


36 


11.474 


71 


22.607 


106 


33-740 


141 


44.890 


37 


II. 791 


72 


22.926 


107 


34-058 


142 


45.204 


38 


12. no 


73 


23.244 


108 


34-376 


143 


45-522 



Table 5 — Pitch Diameters for One-Inch Chordal Pitch 

Teeth from 4 to 14 j 

for any other pitch multiply by that pitch 

CHORDAL THICKNESS OF TEETH 



In order to correctly measure the teeth, the chordal thickness must be 
used, as illustrated by Fig. 40. Also as the location of the pitch line on the 



38 



AMERICAN MACHINIST GEAR BOOK 



sides of the teeth falls below the pitch line at the center of tooth. The 
measurement for the addendum must also be corrected, if any degree of 




FIG, 40. CHORDAL TOOTH THICKNESS. 

accuracy is expected. Table 6, gives these corrected dimensions for various 
standard pitches. 



Number 
of 


I D 


P. 


iVz 


D. P. 


2 D 


. P. 


2^ 


D. P. 


Number 
of 


Teeth. 


Thickness. 


Addendum. 


Thickness. 


Addendum. 


Thickness. 


Addendum. 


Thickness. 


Addendum. 


Teeth. 


8 


1.5607 


1.0769 


1.0405 


0.7179 


0.7804 


0.5385 


0.6243 


0.4308 


8 


9 


1.5628 


1.0684 


1. 0419 


0.7123 


0.7814 


0.5342 


0.6251 


0.4273 


9 


10 


1-5643 


I.0616 


1.0429 


0.7077 


0.7821 


0.5308 


0.6257 


0.4246 


10 


II 


1.5654 


1.0559 


1.0436 


0.7039 


0.7827 


0.5279 


0.6261 


0.4224 


II 


12 


1.5663 


I.0514 


1.0442 


0.7009 


0.7831 


0.5257 


0.6265 


0.4206 


12 


14 


1.5675 


1.0440 


1.0450 


0.6960 


0.7837 


0.5220 


0.6270 


0.4176 


14 


17 


1.5686 


1.0362 


1.0457 


0.6908 


0.7843 


O.5181 


0.6274 


0.4145 


17 


21 


1.5694 


1.0294 


1.0463 


0.6863 


0.7847 


0.5147 


0.6277 


0.4118 


21 


26 


1.5698 


1.0237 


1.0465 


0.6825 


0.7849 


O.5118 


0.6279 


0.4095 


26 


35 


1.5702 


I.0176 


1.0468 


0.6784 


0.7851 


0.5088 


0.6281 


0.4070 


35 


55 


1.5706 


I.OII2 


1. 047 1 


0.6741 


0.7853 


0.5056 


0.6282 


0.4045 


55 


135 


1.5707 


1.0047 


1. 047 1 


0.6698 


0.7853 


0.5023 


0.6283 


0.4019 


135 


Number 
of 


3 D 


. P. 


s'A 


D. P. 


4D 


. P. 


5 E 


>. P. 


Number 
of 


Teeth. 


Thickness. 


Addendum. 


Thickness. 


Addendum. 


Thickness. 


Addendum. 


Thickness. 


Addendum. 


Teeth. 


8 


0.5202 


0.3589 


0.4459 


0.3077 


0.3902 


0.2692 


O.3121 


0.2154 


8 


9 


0.5209 


0.3561 


0.4465 


0.3052 


0.3907 


0.2671 


0.3126 


0.2137 


9 


10 


0.5214 


0.3538 


0.4469 


0.3033 


O.3911 


0.2654 


0.3129 


0.2123 


10 


11 


0.5218 


0.3519 


0.4473 


0.3017 


0.3913 


0.2640 


O.313I 


O.2112 


II 


12 


0.5221 


0.3505 


0.4475 


0.3004 


0.3916 


0.2628 


0.3133 


0.2103 


12 


14 


0.5225 


0.3480 


0.4479 


0.2983 


0.3919 


0.2610 


0.3135 


0.2088 


14 


17 


0.5228 


0.3454 


0.4482 


0.2961 


0.3921 


0.2590 


0.3137 


0.2072 


17 


21 


0.5231 


0.3431 


0.4485 


0.2941 


0.3923 


0.2573 


0.3139 


0.2059 


21 


26 


0.5233 


0.3412 


0.4485 


0.2925 


0.3925 


0.2559 


0.3140 


0.2047 


26 


35 


0.5234 


0.3392 


0.4486 


0.2907 


0.3926 


0.2544 


0.3140 


0.2035 


35 


55 


0.5235 


0.3371 


0.4487 


0.2889 


0.3927 


0.2528 


O.3141 


0.2022 


55 


135 


0.5236 


0.3349 


0.4488 


0.2871 


0.3927 


0.2512 


O.3141 


0.2009 


135 



Table 6 — Chordal Thicknesses and Addenda of Gear Teeth of Diametral Pitch 

Boston Gear Works 



TOOTH PARTS 



39 



Number 
of 


6 D. P. 


7 D. P. 


8 D. P. 


9 D. P. 


Number 
of 


Teeth. 


Thickness. Addendum 


Thickness. Addendum. 


Thickness. 


Addendum. 


Thickness. 


Addendum. 


Teeth, 


8 


0.2601 0.1795 


0.2230 0.1538 


O.1951 


0.1346 


0.1734 


O.1197 


8 


9 


0.2605 0.1781 


0.2233 0.1526 


0.1954 


0.1336 


0.1736 


O.1187 


9 


lO 


0.2607 0.1769 


0.2235 0.1517 


0.1955 


0.1327 


0.1738 


O.1180 


10 


II 


0.2609 0.1760 


0.2236 0.1508 


0.1957 


0.1320 


0.1739 


0.1173 


II 


12 


0.2610 0.1752 


0.2238 0.1502 


0.1958 


0,1314 


0.1740 


0.1168 


12 


14 


0.2612 0.1740 


0.2239 O.1491 


0.1959 


0.1305 


0.1742 


O.1160 


14 


17 


0.2614 0.1727 


0.2241 0.1480 


0.1961 


0,1295 


0.1743 


O.I151 


17 


21 


0.2616 0.1716 


0.2242 0.1471 


0,1962 


0.1287 


0.1744 


O.I 144 


21 


26 


0.2616 0.1706 


0.2243 0.1462 


0.1962 


0.1280 


0.1744 


O.II37 


26 


35 


0.2617 0.1696 


0.2243 0.1454 


0.1963 


0.1272 


0.1745 


O.I 13 I 


35 


55 


0.2618 0.1685 


0.2244 0.1445 


0.1963 


0.1264 


0.1745 


O.I 1 24 


55 


135 


0.2618 0.1675 


0.2244 0.1435 


0.1963 


0.1256 


0.1745 


0.III6 


135 


Number 
of 


10 D. P. 


II D. P. 


12 D. P, 


13 D. P. 


Number 
of 


Teeth. 


Thickness. Addendum. 


Thickness. Addendum. 


Thickness. 


Addendum. 


Thickness. 


Addendum. 


Teeth. 


8 


O.1561 0.1077 


0.1419 0.0979 


O.1301 


0.0897 


0.1201 


0.0828 


8 


9 


0.1563 0.1068 


0.1421 0.0971 


0.1302 


0.0890 


0.1202 


0.0822 


9 


lO 


0.1564 O.I061 


0.1422 0.0965 


0.1304 


0.0885 


0.1203 


0.0816 


10 


II 


0.1565 0.1056 


0.1423 0.0960 


0.1305 


0.0880 


0.1204 


0.0812 


II 


12 


0.1566 0.1051 


0.1424 0.0956 


0.1305 


0.0876 


0.1205 


0.0809 


12 


14 


0.1567 0.1044 


0.1425 0.0949 


0.1306 


0.0870 


0.1206 


0.0803 


14 


17 


0.1569 0.1036 


0.1426 0.0942 


0.1307 


0.0863 


0.1207 


0.0797 


17 


21 


0.1569 0.1029 


0.1427 0.0936 


0.1308 


0.0858 


0.1207 


0.0792 


21 


26 


0.1570 0.1024 


0.1427 0.0931 


0.1308 


0.0853 


0.1207 


0.0787 


26 


35 


0.1570 O.IO18 


0.1427 0.0925 


0.1309 


0.0848 


0.1208 


0.0782 


35 


55 


0.1571 O.IOII 


0.1428 0.0919 


0.1309 


0.0843 


0.1208 


0.0777 


55 


135 


O.I57I 0.1005 


0.1428 0.0913 


0.1309 


0.0837 


0.1208 


0.0772 


135 


Number 
of 


14 D. P. 


15 D. P. 


16 D. P, 


17 D. P. 


Number 
of 


Teeth. 


Thickness. Addendum. 


Thickness. Addendum. 


Thickness. 


Addendum. 


Thickness. 


Addendum. 


Teeth. 


8 


O.1115 0.0769 


0.1040 0.0718 


0.0975 


o!o6 73 


0.0918 


0.0633 


8 


9 


0.1116 0.0763 


0.1042 0.0712 


0.0977 


0.0669 


0.0919 


0.0628 


9 


lO 


O.1117 0.0758 


0.1043 0.0709 


0.0978 


0.0664 


0.0920 


0.0624 


10 


II 


O.1118 0.0754 


0.1044 0.0704 


0.0978 


0.0659 


0.0921 


0.0621 


II 


12 


O.1119 0.0751 


0.1044 0.0701 


0.0979 


0.0657 


0.0921 


0.0618 


12 


14 


O.1119 0.0746 


0.1045 0.0696 


0.0980 


0.0652 


0.0922 


0.0614 


14 


17 


O.1120 0.0740 


0.1046 0.0691 


0.0980 


0.0648 


0.0923 


0.0609 


17 


21 


O.1121 0.0735 


0.1046 o"o686 


0.0981 


0.0643 


0.0923 


0.0605 


21 


26 


O.1121 0.0731 


0.1046 0.0682 


0.0981 


0.0640 


0.0923 


0.0602 


26 


35 


O.1122 0.0727 


0.1047 0.0678 


0.0981 


0.0636 


0.0924 


0.0598 


35 


55 


O.1122 0.0722 


0.1047 0.0674 


0.0981 


0.0632 


0.0924 


0.0595 


55 


135 


0,1122 0.0718 


0.1047 0.0670 


0.0981 


0.0628 


0.0924 


0.0591 


135 



Chordal Thicknesses and Addenda of Gear Teeth oe Diametral Pitch — Continued 

Boston Gear Works 



40 



AMERICAN MACHINIST GEAR BOOK 



Number 
of 


i8 D. P. 


* 19 D. P. 


20 D. P. 


24 D. P. 


Number 
of 


Teeth. 


Thickness. 


Addendum. 


Thickness. 


Addendum. 


Thickness. 


Addendum. 


Thickness. 


Addendum. 


Teeth. 


8 


0.0867 


0.0598 


0.0821 


0.0567 


0.0780 


0.0538 


0.0650 


0.0448 


8 


9 


0.0868 


0-0593 


0.0822 


0.0562 


0.0781 


0.0534 


0.0651 


0.0445 


9 


ID 


0.0869 


0.0589 


0.0823 


0.0558 


0.0782 


0.0530 


0.0651 


0.0443 


10 


II 


0.0869 


0.0586 


0.0824 


0.0555 


0.0783 


0.0528 


0.0652 


0.9439 


II 


12 


0.0870 


0.0584 


0.0824 


0.0553 


0.0784 


0.0525 


0.0653 


0.0437 


12 


14 


0.0871 


0.0580 


0.0825 


0.0549 


0.0784 


0.0522 


0.0653 


0.0435 


14 


17 


0.0871 


0.0575 


0.0826 


0.0545 


0.0784 


0.0518 


0.0653 


0.0432 


17 


21 


0.0872 


0.0572 


0.0826 


0.0542 


0.0785 


0.0514 


0.0654 


0.0429 


21 


26 


0.0872 


0.0568 


0.0826 


0.0538 


0.0785 


0.05 II 


0.0654 


0.0426 


26 


35 


0.0872 


0.0565 


0.0826 


0.0535 


0.0785 


0.0508 


0.0654 


0.0424 


35 


55 


0.0873 


0.0562 


0.0827 


0.0532 


0.0785 


0.0505 


0.0654 


0.0421 


55 


135 


0.0873 


0.0558 


0.0827 


0.0528 


0.0785 


0.0502 


0.0654 


0.0419 


135 



Chordal Thicknesses and Addenda of Gear Teeth of Diametral Pitch — Continued 



Number 
of 


Ys" 


c. p. 


M" 


c. p. 


%" 


c. p. 


1" 


c. p. 


Number 
of 


Teeth. 


Thickness. 


Addendum. 


Thickness. 


Addendum. 


Thickness. 


Addendum. 


Thickness. 


Addendum. 


Teeth. 


8 


0.3105 


0.2142 


0.3725 


0.2570 


0.4347 


0.2997 


0.4968 


0.3426 


8 


9 


0.3109 


0.2125 


0.3730 


0.2550 


0.4353 


0.2976 


0.4974 


0.3400 


9 


10 


O.3112 


0.2II2 


0.3734 


.02534 


0.4357 


0.2957 


0.4978 


0.3378 


10 


II 


O.3114 


0.2100 


0-2,1 ?>7 


0.2520 


0.4360 


0.2941 


0.4982 


0.3360 


II 


12 


0.3116 


0.2091 


0.3739 


0.2510 


0.4363 


0.2938 


0.4986 


0.3346 


12 


14 


O.3118 


0.2077 


0.3741 


0,2492 


0.4366 


0.2908 


0.4988 


0.3322 


14 


17 


0.3120 


0.2061 


0.3744 


0.2473 


0.4369 


0.2886 


0.4992 


0.3298 


17 


21 


0.3122 


0.2048 


0.3746 


0.2457 


0.4371 


0.2868 


0.4994 


0.3276 


21 


26 


0.3123 


0.2036 


0.3748 


0.2443 


0.4372 


0.2851 


0.4997 


0.3258 


26 


35 


0.3124 


0.2024 


0.3748 


0.2429 


0.4373 


0.2833 


0.4999 


0.3238 


35 


55 


0.3124 


0.201 1 


0.3748 


0.2414 


0.4374 


0.2816 


0.4999 


0.3218 


55 


135 


0.3124 


0.1999 


0.5748 


0.2398 


0.4374 


0.2798 


0.4999 


0.3198 


135 


Number 
of 


iVa" 


c. p. 


1V2" 


c. p. 


I^" 


c. p. 


2" ( 


z. p. 


Number 
of 


Teeth. 


Thickness. 


Addendum. 


Thickness. 


Addendum, 


Thickness. 


Addendum. 


Thickness. 


Addendum. 


Teeth. 


8 


0.6210 


0.4284 


0.7450 


0.5140 


0.8694 


0.5994 


0.9936 


0.6852 


8 


9 


0.6218 


0.4250 


0.7460 


0.5100 


0.8706 


0.5952 


0.9948 


0.6800 


9 


10 


0.6224 


0.4224 


0.7468 


0.5068 


0.8714 


0.5914 


0.9956 


0.6756 


10 


II 


0.6228 


0.4200 


0.7474 


0.5040 


0.8720 


0.5882 


0.9964 


0.6720 


II 


12 


0.6232 


0.4182 


0.7478 


0.5020 


0.8726 


0.5876 


0.9972 


0.6692 


12 


14 


0.6236 


0.4154 


0.7482 


0.4984 


0.8732 


0.5816 


0.9976 


0.6644 


14 


17 


0.6240 


0.4122 


0.7488 


0.4946 


0.8738 


0.5772 


0.9984 


0.6596 


17 


21 


0.6244 


0.4096 


0.7492 


0.4914 


0.8742 


0.5736 


0.9988 


0.6552 


21 


26 


0.6246 


0.4072 


0.7496 


0.4886 


0.8744 


0.5702 


0.9994 


0.6516 


26 


35 


0.6248 


0.4048 


0.7498 


0.4858 


0.8746 


0.5666 


0.9998 


0.6476 


35 


55 


0.6250 


0.4022 


0.7499 


0.4828 


0.8748 


0.5632 


0.9999 


0.6436 


55 


135 


0.6250 


0.3998 


0.7499 


0.4796 


0.8748 


0.5596 


0.9999 


0.6396 


135 



Table 7 — Chordal Thicknesses and Addenda of Gear Teeth of Circular Pitch 

Boston Gear Works 



TOOTH PARTS 41 

INVOLUTE CUTTERS 

Until quite recently involute cutters were made in sets of eight, as follows: 



Number 






of Cutter 






I for 


135 teeth to rack 


2 for 


55 to 


134 teeth 


3 for 


35 to 


54 teeth 


4 for 


26 to 


34 teeth 


5 for 


21 to 


26 teeth 


6 for 


17 to 


20 teeth 


7 for 


14 to 


16 teeth 


8 for 


12 to 


13 teeth 



Modern conditions, however, require a more accurate tooth than can be 
produced by this number of cutters. A set of fifteen, utilizing the half 
numbers is now in common use. 

Number 
of Cutter 



Cutter 

1 for 135 teeth to a rack 
13^-2 for 80 to 134 teeth 

2 for 55 to 79 teeth 
23^^ for 42 to 54 teeth 

3 for 35 to 41 teeth 
3H for 30 to 34 teeth 

4 for 26 to 29 teeth 
43-^ for 23 to 25 teeth 

5 for 21 to 22 teeth 
$fy'2 for 19 to 20 teeth 

6 for 17 to 18 teeth 
63^-2 for 15 to 16 teeth 

7 for 14 teeth 
7^^ for 13 teeth 

8 for 12 teeth 



To produce accurate gears, templets for tooth thickness, made according 
to Tables 6 and 7, should be used instead of using one templet for each pitch 
and depending upon the workman's judgment as to how much shake to allow 
for different numbers of teeth. These templets, made up according to Tables 
6 and 7, which are based on the use of eight cutters for each pitch, should be 
sufficiently accurate for all practical purposes. 



SECTION II 



Spur Gear Calculations 



To find the pitch diameters of two gears, the number of teeth in each and 
the distance between centers being given: Divide twice the distance between 
centers by the sum of the number of teeth: Find the pitch diameter of each 
gear separately by multiplying this quotient by its number of teeth. 

Example: Find the pitch diameters of a pair of spur gears, 2 1 and 60 teeth, 
for 25-inch centers. 

2 X 25 



21 + 60 



= 0.617284, 



0.617284 X 21 = 12.96296 inches, or the pitch diameter of the pinion 
0.617284 X 60 = 37.03704 inches, or the pitch diameter of the gear 

The distance between the centers is one-half the sum of the pitch diam- 
eters. In the above example the center distance would prove to be: 

12.96296 + 37.°37°4 ^ ^^ i„^hes 



NOTATIONS FOR FORMULAS 



P 

D' 
D 
V 

d' 
d 

V 

a 
b 



diametral pitch 
pitch diameter 
outside diameter 
velocity 



gear 



pitch diameter 

outside diameter [ pmion 

velocity 

distance between the centers 

number of teeth in both 



These gears run together 



42 



SPUR GEAR CALCULATIONS 



43 



TO FIND 



HAVING 



a and p 
D' and d' 

h and p 
n V and V 

h V and V 

h V and V 

N V and V 

pD' V and v 

N V and n 

pD' V and n 

n V and N 
a V and V 
a V and V 



RULE 



The continued product of center dis- 
tance, pitch and 2 



One-half the sum of the pitch di- 
ameters 



Divide the total number of teeth by 
twice the pitch 



Divide the product of the number of 
teeth and velocity of pinion by the 
velocity of gear 



Divide the product of the total num- 
ber of teeth and velocity of pinion 
by the sum of the velocities 



Divide the product of the total num- 
ber of teeth and the velocity of gear 
by the sum of the velocities 



Divide the product of the number of 
teeth in gear and its velocity by 
the velocity of pinion 



Divide the continued product of the 
pitch, pitch diameter and velocity 
of the gear by the velocity of pinion 

Divide the product of the number of 
teeth and velocity of gear by the 
number of teeth in pinion 



Divide the continued product of the 
pitch, pitch diameter and velocity 
of gear by the number of teeth in 
pinion 



Divide the product of the number of 
teeth in pinion and its velocity by 
the number of teeth in gear 



Divide the continued product of the 
center distance, velocity of pinion 
and 2, by the sum of the velocities. 

Divide the continued product of the 
center distance, velocity of gear and 
2, by the sum of the velocities 



FORMULA 



a p 2 

D' + d' 
2 

b 

2p 

n V 



b V 

V+v 

v+ V 

NV 

V 

pD' V 



NV 



pD'V 



n V 



N 



2 a V 
v+ V 

2^aV 
v+ V 



EXAMPLE 



15 X 3 X 2 =90 
20 + 10 

2 •>' 

90 



2 X3 ^5 

30 X 2 



90 X 2 
2 + 1 

90 X I 

2 + 1 

60 X I 

2 

3 X 20 X I 



= 60 



= 60 



= 30 



= 30 



= 30 



60 X I 

30 



= 2 



3 X 20 X I 





30 






30 X 2 






60 




2 


X 15 X 

2 + 1 


2 


2 


X 15X 


I 



= 2 



2 + 1 



= I 



= 20 



= 10 



Table 8 — Formulas for a Pair of Mating Spur Gears 



44 



AMERICAN MACHINIST GEAR BOOK 



TO FIND 



The Diametral 
Pitch. 

The Diametral 
Pitch. 

The Diametral 
Pitch. 



Pitch 



Diameter. 



Pitch 



Diameter. 



Pitch 



Diameter. 

Pitch 

Diameter. 

Outside 

Diameter. 

Outside 

Diameter, 

Outside 

Diameter, 

Outside 

Diameter, 

Number of 

Teeth, 

Number of 

Teeth, 

Thickness 

of Tooth, 

Addendum. 



Dedendum. 
Working 

Depth, 

Whole Depth, 

Clearance. 
Clearance. 



HAVING 



The Circular Pitch. 

The Pitch Diameter 
and the Number of 
Teeth 

The Outside Diame- 
ter and the Number 
of Teeth 

The Number of Teeth 
and the Diametral 
Pitch 

The Number of Teeth 
and Outside Diam 
eter 

The Outside Diame- 
ter and the Diame 
tral Pitch 

Addendum and the 
Number of Teeth . . 

The Number of Teeth 
and the Diametral 
Pitch 

The Pitch Diameter 
and the Diametral 
Pitch 

The Pitch Diameter 
and the Number of 
Teeth 

The Number of Teeth 
and Addendum . . 

The Pitch Diameter 
and the Diametral 
Pitch 

The Outside Diame- 
ter and the Diame- 
tral Pitch 



The Diametral Pitch. 
The Diametral Pitch. 

The Diametral Pitch. 
The Diametral Pitch. 
The Diametral Pitch. 

The Diametral Pitch. 
Thickness of Tooth.. 



RULE 



Divide 3.1416 by the Circular 
Pitch 



Divide Number of Teeth by 
Pitch Diameter 



Divide Number of Teeth plus 2 
by Outside Diameter 



Divide Number of Teeth by the 
Diametral Pitch 



Divide the product of Outside 
Diameter and Number of Teeth 
by Number of Teeth plus 2 . . . 

Subtract from the Outside Diam- 
ter the quotient of 2 divided by 
the Diametral Pitch 

Multiply Addendum by the 
Number of Teeth 



Divide Number of Teeth plus 2 
by the Diametral Pitch 



Add to the Pitch Diameter the 
quotient of 2 divided by the 
Diametral Pitch 

Divide the product of the Pitch 
Diameter and Number of Teeth 
plus 2 by the Number of Teeth 

Multiply the Number of Teeth 
plus 2 by Addendum 



Multiply Pitch Diameter by the 
Diametral Pitch 



Multiply Outside Diameter by 
the Diametral Pitch and sub- 
tract 2 

Divide 1.5708 by the Diametral 
Pitch 

Divide i by the Diametral Pitch, 
D' 



or s = 



N 



Divide 1.157 by the Diametral 
Pitch 



Divide 2 by the Diametral Pitch. 

Divide 2.157 by the Diametral 
Pitch 

Divide 0.157 by the Diametral 
Pitch 

Divide Thickness of Tooth at 
pitch line by 10 



FORMULA 



3.1416 




N + 2 



D' ^ D - — 
D' = sN 

N+2 



D = 



D-D' + — 



D = 



{N+ 2) ly 



N 
D = {N+2) s 

N = D' p 

N = Dp- 2 

^= ^:57o8_ 



5 = 



s+f 



W = 



I-I57 



W'+f 



2.157 



f = 



0-157 
P 



f--^ 



Table 9 — Spur Gear Calculations for Diametral Pitch 

14^^ Degree Standard 

|2. D. Nuttall Company 



SPUR GEAR CALCULATIONS 



45 



TO FIND 



The Circular 
Pitch 

The Circular 
Pitch, 

The Circular 
Pitch. 



Pitch 



Diameter. 



Pitch 



Diameter. 



Pitch 



Diameter. 

Pitch 

Diameter. 

Outside 

Diameter. 

Outside 

Diameter. 

Outside 

Diameter. 

Number of 

Teeth. 

Thickness 

of Tooth. 

Addendum. 



Dedendum. 

Working 

Depth. 

Whole Depth. 

Clearance. 

Clearance. 



HAVING 



RULE 



The Diametral Pitch. 

The Pitch Diameter 
and the Number of 
Teeth 

The Outside Diame- 
ter and the Number! 
of Teeth 

The Number of Teeth 
and the Circular 
Pitch 



Divide 3.1416 by the Diametral 
Pitch 



Divide Pitch Diameter by the 
product of 0.3183 and Number 
of Teeth 



The Number of Teeth 
and the Outside Di- 
ameter 



The Outside Diame- 
ter and the Circular 
Pitch 



Addendum and the 
Number of Teeth. . 

The Number of Teeth 
and the Circular 
Pitch 



The Pitch Diameter 
and the Circular 
Pitch 



The Number of Teeth 
and the Addendum 

The Pitch Diameter 
and the Circular 
Pitch 



The Circular Pitch. . . 
The Circular Pitch. . . 

The Circular Pitch. . . 
The Circular Pitch. . . 
The Circular Pitch. . . 
The Circular Pitch. . . 
Thickness of Tooth.. 



Divide Outside Diameter by the 
product of 0.3183 and Number 
of Teeth plus 2 

The continued product of the 
Number of Teeth, the Circular 
Pitch and 0.3183 

Divide the product of Number of 
Teeth and Outside Diameter 
b}' Number of Teeth plus 2 . . 

Subtract from the Outside Diam 
eter the product of the Circular 
Pitch and 0.6366 

Multiply the Number of Teeth by 
the Addendum 



FORMULA 



/»' = 
p' = 

p'=. 



3-1416 

p 

D' 

0.3183 iV 

D 



0.3183 iV+2 

U = i\^/ 0.3183 



The continued product of the 
Number of Teeth plus 2 the 
Circular Pitch and 0.3183 ... . 

Add to the Pitch Diameter the 
product of the Circular Pitch 
and 0.6366 

Multiply Addendum by Number 
of Teeth plus 2 

Divide the product of Pitch Di- 
ameter and 3.1416 by the Cir 
cular Pitch 



D' = 



ND 

N + 2 



One-half the Circular Pitch , 



Multiply the Circular Pitch by 

D' 

0.3183, or s = 



N 



Multiply the Circular Pitch by 
0.3683 

Multiply the Circular Pitch by 
0.6366 

Multiply the Circular Pitch by 
0.6866 

Multiply the Circular Pitch by 
0.05 

One-tenth the Thickness of Tooth 
at Pitch Line 



D' = D- 

ip' 0.6366) 

D' = N 5 

D = {N+2) 

/ 0.3183 

D = D + 

ip' 0.6366) 

D = s{N + 2) 

Z>'3.i4i6 



N = 



t = 



P' 



P' 



s = /)' 0.3183 

5-f/=/o.3683 
W = p' 0.6366 
W = p' 0.6866 
f = p 0.05 
t 



/ = 



10 



Table 10 — Spur Gear Calculations for Circular Pitch 

M/^ Degree Standard 

.^, v. Nuttall Company 



SECTION III 

Speeds and Powers 

transmission of power by gearing with particular reference to 

spur and bevel gears 

SPEED RATIO 

The problem of finding the proper diameter or speed of a gear or pulley is 
simple enough when once thoroughly understood. 

The gear may be represented by its number of teeth, pitch diameter, pitch 
radius, or speed ratio, as the case may be. In the explanation to follow the 
number of teeth is used. The speed is in revolutions per minute. 

Rule: Divide the product of the speed and number of teeth of one gear by 
the speed or number of teeth of its mate to secure the lacking dimension. 

That is, if both the speed and number of teeth are known for one gear, 
multiply the speed by the number of teeth, and divide this product by the 
known quantity of the mating gear to secure its number of teeth or speed, as 
the case may be. 

Or the same result may be obtained by proportion, the values being 
placed as follows : 

n : N : R :r (i) 

n = number of teeth in pinion 
r = revolutions per minute of pinion 
N = number of teeth in gear 
R = revolutions per minute of gear 

Example: A gear having 60 teeth makes 300 revolutions per minute, what 
will be the speed of an engaging pinion having 15 teeth? 

n : N : R :r 

15 : 60 : 300 : x 

Therefore, x = — ? or 1,200 revolutions per minute for pinion n 

To compute these values for a train of gears, use the continued product of 
the pinions and the continued product of the gears as a single gear and pinion 
and proceed as above. 

Example: In Fig. 41, the gear N has 100 teeth, N', 70 teeth, .Y", 60 
teeth, w, 15 teeth; n\ 18 teeth; a,ndn" , 24 teeth. The gear A" makes 10 revolu- 
tions per minute. What will be the speed of the pinion n'^? 

46 



SPEEDS AND POWERS 47 

N, N' and N'' = 100 X 70 X 60 = 420,000 

n, n' and n" = 15X18X14 = 6,480. 

w : iV : : i? : r 

6,480 : 420,000 :: 10 '. x 

rx., r 420,000 X 10 . „ , . . . . . ,, 

Therefore, x = — ^ 7 or 648 revolutions per minute for pinion n . 

6,480 




FIG. 41. GEAR TRAIN. FIG. 42. INTERMEDIATE GEAR DOES 

NOT AFFECT THE SPEED RATIO. 

The velocities of a train of gears may also be found as follows: N, N', N" 
and n, n' , n" , etc., representing the number of teeth in the gears and pinions. 

_ RNN'N" 

'' ~ nn' n"' ^^^ 

_ _ r n n' n" , . 

NN'N'' ' ^^^ 

The intermediate gear B, as shown in Fig. 42, while it changes the direc- 
tion of the rotation of the gears, A and C does not alter their speed ratio, the 
circumferential velocities of all three gears being equal. 



ARRANGEMENT OF GEAR TRAINS 

For compound reduction there must be four gears, as per Fig. 43, the 
gears B and C being keyed to an intermediate shaft, the power being trans- 
ferred to the machine by the shaft-carrying gear D. 

When a great reduction is required, say 64 to i, there may be two inter- 
mediate shafts, as in Fig. 44. 

This reduction might be accomplished by using a drive, as in Fig. 43, 
dividing the total reduction between two sets of gears, but a triple reduction 
is used by way of illustration. The best results are always obtained by 
dividing the reduction as evenly as possible among the different pairs of 
gears. For instance: For a double reduction, as in Fig. 43, the ratio of each 
pair should be made as near the square root of the total reduction as possible. 
In case of the triple reduction. Fig. 44, the ratio of each pair should be the 
cube root of the total reduction, or \/64 = 4. That is, there are three sets of 
gears, each having a speed ratio of 4 to i. If double reduction had been used 
the reduction of each gear would have been ^764 = 8, or two sets of gears 
each having a speed ratio of 8 to i. 

Gear trains proportioned in this way give the highest possible efficiency. 
For instance: An unsuccessful single gear reduction of 16 to i might be 



48 



AMERICAN MACHINIST GEAR BOOK 



made efficient by substituting two pairs of gears, each having a ratio of 4 to i. 
Making the compound gears 8 to i and 2 to i would help, but would not be as 



B 



I 



Motor 



-P\P\ 



WW 



FIG. 43. DOUBLE GEAR 
REDUCTION. 



B 



-4 



Motor 



y w 



FIG. 44. TRIPLE GEAR 
REDUCTION. 



efficient as the equal reduction. This will be especially noticeable in long 
leads in the lathe or milling machines. 



POWER RATIO 



The relative powers of a train of gears are inversely proportional to their 
circumferential velocities. The circumferential velocity of each pair of gears 




FIG. 45. FIG. 46. 

POWER RATIO DIAGRAMS. 



in a train being equal, the driving pinion, as shown in Figs. 45 and 46, is 
ignored in the calculations for a single pair, the circumferential velocity and 
the load on the teeth being the same as for the mating gear. The problem is 
to determine the power ratio between the drum r and the gear R. 



SPEEDS AND POWERS 49 

Ignoring friction, the values of this drive may be found by proportion, 
arranged as follows: 

W :R::F :r (4) 

Enough must be added to the load W or taken from the effective lifting 
force F to overcome the f rictional resistance of the teeth and bearings. This 
loss must be estimated and the percentage of loss added to the load W , the 
ratio of R and r being determined according to this new ratio. 

Example: Referring to Fig. 45: If the radius of the gear R is 18 inches, the 
radius of the drum r three inches, what power will be required at F to raise 
300 pounds at W? 

W :F ::F :r 

300 : 18 : : .T : 3 
Therefore, x = — -j = 50 pounds required at F. 

Suppose the loss in efificiency to be 10 per cent and the radius of the gear R 
18 inches. What must be the radius of the drum r to raise 300 pounds at IF ? 

300 + 10 per cent = 330 pounds. 
W :R::F :T 

300 : 18 : : 50 : X 

Therefore, x = —^ = 2.7 inches for the radius of drum r. 

330 

For a train of gears, the continued products of the driving and driven gears 
may be considered as single gears. Or the power ratio may be considered 
between each pair inversely proportional to their velocity ratios. 

Example: Referring to Fig. 47; what force is required at F to raise 2,500 
pounds at W , the loss in efficiency being 30 per cent? 

R R' R" = 20 X 18 X 10 = 3,600 
y rV'' = 6 X 8 X 5 = 240 

W = 2,500 + 30 per cent = 3,250 
W :R : : 

3,250 : 3,600 : : X : 240, x = F : r or 217 pounds at F. 

., ^ Wrr'r" 3,250X6X8X5 a r\ 

^^^^ ^ = RR'~W^ = 20 X 18 X 10 = ''^ P"""^^- ^5) 

. , __. FRR'R" 217 X 20 X 18 X 10 , ,.. 

And W = -77,-^, = 5v^3;^^ ' = 3,250 pounds. (6) 

AN EXAMPLE IN HOIST GEARING 

Example: What gears will be required to lift a load of 2,400 pounds at a 
uniform rate of speed, employing a 10 horse-power motor running 1,120 
4 



50 



AMERICAN MACHINIST GEAR BOOK 



revolutions per minute, driving with a rawhide pinion 4 inches pitch diam- 
eter? See Fig. 48. 

F = 281 Pounds 




1120 
R.P.M 




1173 Feet 
per Min. 



2400 Pounds- 



FIG. 47. POWER RATIO OF GEAR TRAINS. FIG. 48. EXAMPLE OF GEAR DRIVE FOR HOIST. 



Velocity of pinion in feet per minute, V = d' 0.261S R. P. M. 

HP X 33,000 



The safe load, W + 



V 



(7) 
(8) 



Therefore, F = 4 X 0.2618 X 1,120 =1,173 ^^^^ per minute. 

T O ^^ '2 '2 000 

And W = ^^ ' or 281 pounds, which is the load to be carried by 

1,173 
the pinion. 

Assuming that 20 per cent is lost by the friction of the gear teeth, bearings, 

etc., the real load to be raised by the force of 280 pounds at the pitch line of 

the driving pinion is: t 

2,400 + 20 per cent. = 2,880 pounds. 

The necessary velocity ratio of the gears to equal this ratio of power is, 
therefore : 

2,880 _ 10.25 
^281"' ~ I 

This reduction must be made between R' and r", and R and r', the pinion 
r not being considered as its velocity is the same as that of the gear R, there- 
fore the load on the teeth will be the same. 

Since it is always best to make the reduction in even steps, and double 
reduction is desirable for a ratio of 10.25 to i take the square root of the total 
reduction, 10.25, which is approximately 3.2 to i for each reduction. Prac- 
tically, however, a reduction of and - will answer. 

The ratio between R' and r" is made ~ — Assuming the diameter of the 

drum r" to be 10 inches, the pitch diameter of the gear R' will be 3.4 X 10 = 

34 inches. The ratio between R and / is . Assuming the pitch diameter of 

the pinion r' to be 7 inches, the pitch diameter of the gear R will be 7 X 3 = 
21 inches. 



SPEEDS AND POWERS 51 

The power or circumferential force of the gear R is, of course, that of the 
driving pinion r, 281 pounds. Therefore, the power of the pinion r', and 
consequently that of the gear R', is 281 X 3 = 843 pounds. 

The problem is now reduced to two simple ones, that is — to design a pair of 
gears r and R to transmit a force of 281 pounds at a speed of 1,173 ^^^t per 
minute, and a second pair r' and R' to transmit a force of 843 pounds at a 
speed of 390 feet per minute. 

It is necessary to assume a pitch judged to be suitable for the different 
drives and to try its value for carrying the required load by the Lewis formula, 
obtaining the safe load per inch of face, and make the face sufficiently wide to 
transmit the power. 

For the first pair of gears, r and R, assume 4 diametral pitch — 0.7854-inch 
circular pitch — allowing 5000 pounds per square inch as a safe stress for raw- 
hide. Number of teeth in pinion ^ = 4X4= 16. 

Safe load per inch of face = spy -r- — , — f>- (See formula 24.) 

000 + V 

Or s,ooo X 0.78=: X 0.0777 ■ = 100 pounds per inch of face. 

^ ' ^ " 600 + 1,173 

Making the face of the gears r and R 3 inches it will safely carry 3 X 100 
= 300 pounds, which is sufficient. 

For the second pair, r' and R', try 3 diametral pitch — 1.0472-inch circular 
pitch — both gears of cast iron. Figure the strength of the pinion, as it is the 
weaker of the two. Allow 8,000 pounds per square inch as a safe stress. 

For a pinion 7 inches pitch diameter, 3 pitch, the number of teeth equals 

7X3 = 21 teeth. Factor y for 21 teeth equals 0.092. W = 8,000 X 1.047 

600 
X 0.092 7 ^r y or 467 pounds per mch of face. 

Making the face 3 inches, the gears will carry a load of 3 X 467 =1,401 
pounds. These gears will therefore be heavier than necessary, but owing to 
the nature of the service this should be the case, especially as they are made 
of cast iron. 

From the ratio of this train of gears it will be found that the load will be 

I 17s 
raised at — '— =115 feet per minute, using the full speed of the motor. 

If the load must be raised at a greater speed than 115 feet, a more powerful 
motor would be required, and if at a lower speed there must be a greater gear 
reduction. For instance, if the hoisting speed had been 80 feet per minute 

.1 1 • 111 i»i7S 14-7 • 1 r IO-2 

the speed ratio would be -^ — = ' instead of as m the example. 

80 I I ^ 

The above problem is generally put before the designer in a different 
manner — that is, the load and speed at which the load is to be raised are 
given, the size of motor and ratio of gearing, etc., to be determined. 

Example: A load of 2400 pounds is to be raised at the uniform rate of 115 
feet per minute; what size motor and what gears will be required? 



52 



AMERICAN MACHINIST GEAR BOOK 



Assuming as before a loss of 20 per cent in efficiency in the driven gears, 
bearing, etc., this load will require: 

-f = 10 horse-power (2,400 pounds + 20 per cent = 2,880 pounds). 

33^000 

Using a rawhide pinion 4-inch pitch diameter on the motor, we con- 
sider the problem in the same manner as in previous examples making the 

2,880 10.25 
ratio of the gears; —^ — 



ii I 

The problem of determining the proper gears is the same. 

RAILWAY GEARS 



Speed in feet per minute at rim of car wheel V = 

miles per hour. 
Speed in feet per minute at pitch line of gear V = 



X speed of car in 

(9) 

X miles per hour 

(10) 



Ratio of gear to wheel R = 

Force at pitch line of gear 

Fiber stress in tooth 6* = 

Traction effort at wheel T = 

Horse power HP = 

Speed of car in miles per hour M = 



pitch diameter of gear 

diameter of wheel 
P HP X 33,000 Kw X 44jI02 



V 
F 



V 



(11) 
(12) 



P'fy 

F 
R 

T M 



600 



600 -I- V 



(See formula 18.) (13) 

(14) 



Traction effort 



T = 



0-375 

Dia. of wheel X teeth in pinion X 

revolution per minute of pinion 

teeth in gear X 336 
teeth in gear X 24 X gear efficiency 
X torque of motor 
M X diameter of wheel 



(15) 



(16) 



(17) 



F or Pressure at Pitch 
Line =3553 Pounds. 




Tractive Effort = 5040 Pounds. 

Prom Formula 12, F= 5040 x 26.8 —^^q Pounds 
38 

FIG. 49. RAILWAY GEARS. 



Example: A car weighing 60 tons 
driven by four motors accelerating at 
the rate of i}-^ miles per hour, per 
second, reaches the peak of its start- 
ing torque when at a speed of 20 
miles per hour. The gears are 20 and 
67 teeth 23^^ diametral pitch (1.26 
inches circular pitch) 5J^-inch face. 
The diameter of the car wheels is 38 
inches. It is required to know the 
maximum fiber stress in pinion tooth. 
The power exerted by motors at its 
peak is 400 kilowatts (800 amperers at 
500 volts). See Fig. 49. 



SPEEDS AND POWERS 



S3 



Kilowatts per motor 
Pitch diameter of gear 
Ratio of gear to wheel 



Kw = ^^— = loo Kw 



D' = -% = 26.8 in. 

T, 26.8 

R = '^^ = 0.705 



3^ 



Speed of gear in feet per minute 

F = 88 X 20 X 0.705 = 1,241 feet per minute. 



Force at pitch line 

Fiber stress in pinion tooth 



5 = 



3^553 



1.26 X 5.25 X 0.09 X 



„ 100 X 44>io2 

F = 7777-^ = 3,553 pounds. 

1,241 

- = 18,300 pounds per square inch. 



600 



600 + 1,241 



ST-RENGTH OF GEAR TEETH 

Lewis 

W = load transmitted in pounds (same as value F), 

p' = circular pitch, 

/ = face, 

y = factor for different numbers and forms of teeth (Table 11), 

S = safe working stress of material, 

V = velocity in feet per minute, 

600 



W = Sp'fy 



60 + F 



(18) 



NUMBER 


VALUE OF FACTOR y 


NUMBER 


VALUE OF FACTOR y 














OF 
TEETH 


INVOLUTE 
20° 


INVOLUTE 

15° 
CYCLOID AL 


RADIAL 
FLANKS 


OF 
TEETH 


INVOLUTE 
20° 


INVOLUTE 

15° 
CYCLOID AL 


RADIAL 
FLANKS 


12 


0.078 


0.067 


0.052 


27 


O.III 


O.IOO 


0.064 


13 


0.083 


0.070 


0-053 


30 


0.II4 


0.102 


0.065 


14 


0.088 


0.072 


0.054 


34 


0.II8 


0.104 


0.066 


IS 


0.092 


0.075 


0.055 


38 


0.122 


0.107 


0.067 


16 


0.094 


0.077 


0.056 


43 


0.126 


O.IIO 


0.068 


17 


0.096 


0.080 


0.057 


50 


0.130 


O.II2 


0.069 


18 


0.098 


0.083 


0.058 


60 


0.134 


O.II4 


0.070 


19 


O.IOO 


0.087 


0.059 


75 


0.138 


O.I16 


0.071 


20 


0.102 


0.090 


0.060 


100 


0.142 


O.I18 


0.072 


21 


0.104 


0.092 


0.061 


150 


0.146 


0.120 


0.073 


23 


0.106 


0.094 


0.062 


300 


0.150 


0.122 


0.074 


25 


0.108 


0.097 


0.063 


Rack 


0.154 


0.124 


0.07s 



Table ii — Values of Factor y for Lewis Formula 

Safe working stress S for 0.30 carbon steel = 15,000 

Safe working stress 5* for 0.50 carbon steel = 25,000 

Safe working stress 5 for cast iron = 8,000 

Safe working stress S for rawhide = 5,000 



54 



AMERICAN MACHINIST GEAR BOOK 



AVERAGE VALUES FOR 5 

Mr. Lewis' formula for the strength of gears originally read: W = S p^fy^ 
a table being given in which the allowable stress of the material S was reduced 
as the speed of the gear was increased as follows : 



SPEED OF TEETH IN 
FEET PER MINUTE 


lOO OR 
LESS 


200 


300 


600 


900 


1200 


1800 


2400 


Cast Iron 

Steel 


8,000 
20,000 


6,000 
15,000 


4,800 
12,000 


4,000 
10,000 


3,000 
7,500 


2,400 
6,000 


2,000 
5,000 


1,700 
4,300 







Safe Working Stresses S in Pounds Per Square Inch for Different 

Speeds 



Later Carl G. Barth introduced an equation, 



600 



., which gives prac- 



600 + F' 

tically the same result as the table when added to the formula, the value S 
being the safe working stress per square inch of the material used, or 



W = Spjy 



600 



600 + V 

Mr. Earth's equation is the one commonly accepted. It is evident, 
however, that this value will vary for different conditions, the design and 
workmanship being important factors in its proper determination. 

The load is reduced as the speed increases on account of impact. It is 
evident that an accurately spaced and generated gear should have a much 
higher value than one cut by ordinary methods. 

It is also evident that helical and herringbone gears, owing to the nature of 
their tooth contact, should have a much higher value, as they operate under 
entirely different conditions, therefore are capable of heavier loads at higher 
speeds for the same area of tooth contact. Rawhide gears should also have 
a higher factor, as rawhide will absorb shocks that would affect harder 
materials. 

In the absence of all vibration, and with an indeflectable material, this 
equation could be eliminated from the formula for strength and wear. These 
are conditions that can never be attained, but it is evident that this value 
will stand extended investigation. 



STRENGTH OF BEVEL GEARS 



In general apply the Lewis formula for spur gears, figuring the safe load 
from the average pitch diameter, or, stated a little differently, the velocity in 
feet per minute and the pitch is to be taken at the average pitch diameter, 
otherwise the gear is to be treated as a spur. 



SPEEDS AND POWERS 



55 



Let N = number of teeth. 

p' = circular pitch. 

D' = pitch diameter. 

h = face width. 

p'a = average circular pitch. 

Da = average pitch diameter. 

a = apex distance. 

E = center angle. 

S = safe working stress. 

Va = velocity (average) in feet per minute. 

In order to get the average pitch we must first determine the apex distance 

a. Now 

D' 

2 sin. h 

The average pitch is the pitch at the center of the gear face/. Above this 

section the tooth strength increases and below this point it decreases. The 



s 




-^ 




FIG. 50. DIAGRAM FOR STRENGTH OF BEVEL GEARS. 

mean strength of the tooth is, therefore, located at this point. Thus it is the 
proper dimension to use in determining the strength of the tooth. This 
average pitch is found from the following equation: 

^ a 



56 AMERICAN MACHINIST GEAR BOOK 

This formula is derived as follows: By referring to Fig. 51, it is evident 
that the pitch of the tooth at p' is to the apex distance a as the pitch at p' a is 

to the mean apex distance ia — -y 

The average pitch diameter Da is found from formula (3) by substituting 
the average pitch. The average velocity Fa is found from formula (4) by 
using the average pitch diameter. Then 

- = -^- M 

2 stn E 

K« - D 

P'a = \ " ■ {2) 

(m 

Da = Np'a 0.3183. (3) 

Va = o.26i8Z)«(r. p. m.). (4) 

Safe load = 5/„*,(^-^^j. (5) 

TT Safe load X Va ,^. 

Horse-power = • (6) 

33,000 

The values for the factor y are from the Lewis formula. 
An illustrative example is as follows: What power may be transmitted 
by a pair of miter gears of the following dimensions: 30 teeth, 2-inch pitch, 
5-inch face, 19.107-inch pitch diameter, at 50 revolutions per minute? The 
material is cast iron. 

a = , — - — = i^.i;i inches, the apex distance; 
2 X 0.707 ^ ^ ' F 

p'a= ^^ = 1.63 inches, the average pitch; 

^a= 30 X 1.63 X 0.3183 = 15.5 inches, the average pitch 

diameter; 
V a= 0.2618 X 15.5 X 50 = 203, the average velocity in feet per minute; 

The safe load = 8,000 X i.6s X S X 0.102 X (7 , I = 4,070 

^ "^ \6oo + 203/ ^' 

pounds. 

rr^i 1 4r97o X 203 

The horse-power = -^^ = SO-S- 

^ 33,000 

The teeth in bevel gears are more strongly shaped than the teeth of spur 
gears of the same pitch and number. This increase is represented by the 
radius e — p' a in Fig. 51, compared with the radius at the point p'. The 
corresponding number of teeth for this larger radius is found by the expression 

N 
cos E 

When selecting the constant y, however, it is well to disregard this increase, as 
it will tend to compensate for the loss in efficiency due to the use of bevel gears. 



SPEEDS AND POWERS 57 

NOTE ON FORMULAS FOR THE STRENGTH OF GEAR TEETH 

The Lewis formula is not directly applicable to all types of gears, unless 
they are of the straight tooth variety with teeth of standard proportions. 
The absence of shock — sudden transference of load between successive teeth — 
in certain types of gears and the axial thrust introduced by an oblique 
arrangement of teeth bring in conditions which necessitate the modification of 
the original formula as corrected by Barth and as adapted to bevel gearing. 
W. C. Bates found that the transference of load from tooth to tooth without 
shock in herringbone gearing modified the speed factor ratio added to the 
Lewis formula by Barth by changing the constant 600 to 1,200. 

In the case of spiral gears, and also helical gears, the same modification 
of the formula is necessary, as these types are similar so far as elimination of 
sudden appHcation of load is concerned. A second modification is necessary 
for gears with obliquely arranged teeth, as the total load on any type of spiral 
gear tooth is the resultant of the transmitted load and the thrust and is 
equal to the transmitted load multiplied by the secant of the spiral angle. 
The safe load is then the load which could be carried were it not for the thrust 
multiplied by the cosine of the spiral angle. 

As the factor y for the Lewis formula, given in Table 11, is based upon the 
condition that the proportions of the teeth conform to standard still another 
modification in this factor is necessary when employing the Lewis 
factor to gears with long or short addenda. In the case of modified teeth 
(see page 26), the breaking moment arm is altered and the thickness of the 
teeth at the top of the root fillets is affected. 

These various modifications of the original Lewis formula convert that 
equation into a series of formulas, as follows, the strength factors for which 
are given in Table 12. 

MODIFIED LEWIS FORMULAS 
STANDARD ADDENDUM, 

Straight Tooth Spur Gears Safe Load = Sp'Jy - — -r-^ 

Herringbone Gear Safe Load = Sp'Jy 



1,200 + V 

1,200 



Helical and Spiral Gears Safe Load = Sp'fy cos A . ^. 

^ •^'^ 1,200 + V 

Straight Tooth Bevel Gear Safe Load = Sp"fy , y, 

Herringbone Bevel Gear Safe Load = Sp'Jy . „; 

Helical and Spiral Types BevelGears Sa.ieL.oa,d = Sp"fy cos A 



1,200 + F' 



58 AMERICAN MACHINIST GEAR BOOK 

SHORT ADDENDUM (0.3 X working depth of tooth) 

Straight Tooth Spur Gears Safe Load = Sp'fy' , ^— p 

Herringbone Gear Safe Load = Sp'fy' 



Helical and Spiral Gears Safe Load = Sp'fy' cos A 



1,200 + V 

1,200 



1,200 + V 



LONG ADDENDUM (0.7 X working depth of tooth) 

Straight Tooth Spur Gears Safe Load = Sp'fy" - — ^jZ~v 

Herringbone Gear Safe Load = Sp'fy" 



1,200 + V 

Helical and Spiral Gears Safe Load = Sp'fy" cos A ' — , — -. 

^ '^ 1,200 -j- V 

SHORT ADDENDUM (0.3 X working depth of tooth) 

Straight Tooth Bevel Gear Safe Load = Sp"fy' x^' 

Herringbone Bevel Gear Safe Load = Sp"fy' . ,, / 

Helical and Spiral Types Bevel Gears Safe Load = Sp"fy" cos A - — ^ — r-777 
^ ^ '^ -^ I,200+K 

LONG ADDENDUM (0.7 X working depth of tooth) 

Straight Tooth Bevel Gear Safe Load = Sp"fy" , 

Herringbone Bevel Gear Safe Load = Sp"fy" '- — v-^rr, 

■^ 1,200 + V 

Helical and Spiral Types Bevel Gears Safe Load = Sp"fy" cos '- — , — jy) 

^ 1,200 + V 

Notation: 

Safe working stress (pounds per square inch) S 

Circular pitch p' 

Average circular pitch (bevel gears) p" 

Face of gear / 

Pitch circle velocity (feet per minute) V 

Average pitch circle velocity — bevel gears (ft. per minute.) V 

Strength factor — standard addendum y 

balanced thrust, short addendum y' 

balanced thrust, long addendum y" 

unbalanced thrust, short addendum Y' 

unbalanced thrust, long addendum Y" 

Spiral angle — helical and spiral gears, unbalanced thrust A 



SPEEDS AND POWERS 59 

Table 12. — Strength Factors for Modified Lewis Formulas 



NUMBER 




VALUES OF FACTORS 




NUMBER 


OF 












OF 


TEETH 












TEETH 




y 


y 


y" 


Y' 


Y" 




8 


0.052 


0.051 


0.062 


0.053 


0. Ill 


8 


9 


0-055 


0.054 


0.066 


0.058 


0. 127 


9 


10 


0.059 


0.056 


0.070 


0.064 


0. 142 


10 


II 


0.063 


0.059 


0.074 


0.069 


0.157 


II 


12 


0.067 


0.062 


0.078 


0.076 


0. 172 


12 


13 


0.070 


0.063 


0.081 


0.079 


0.185 


13 


14 


0.072 


0.065 


0.083 


0.082 


0.195 


14 


IS 


0.075 


0.067 


0.086 


0.086 


0. 208 


15 


16 


0.077 


0.069 


0.089 


0.089 


0. 216 


16 


17 


0.080 


0.071 


0.092 


0.093 


0. 227 


17 


18 


0.083 


0.073 


0.095 


0.097 


0.238 


18 


19 


0.087 


0.076 


0.099 


0. lOI 


0.251 


19 


20 


0.090 


0.079 


0.103 


0.105 


0.263 


20 


21 


0.092 


0.080 


0.105 


0. 108 


0. 272 


21 


23 


0.094 


0.082 


0. 108 


0. Ill 


0. 282 


23 


25 


0.097 


0.084 


0. Ill 


0.115 


0.294 


25 


27 


0. 100 


0.086 


0. 114 






27 


30 


0. 102 


0.088 


0. 117 






30 


34 


0. 104 


0.089 


0. 120 






34 


38 


0. 107 


0.091 


0. 122 






38 


43 


0. no 


0.093 


0. 126 






43 


50 


0. 112 


0.095 


0.128 






50 


60 


0. 114 


0.096 


0.130 






60 


75 


0. 116 


0.097 


0.132 






75 


100 


O.I18 


0.099 


0.134 






100 



FACTOR OF SAFETY FOR GEARS 

The load on the teeth of gears made from f orgings may be such as to strina 
the material close to its elastic limit (based upon the worn thickness of tooth), 
if it is free from flaws. For castings this is not a safe rule, as there are always 
hidden defects to a greater or less extent. As long as the strain is kept below 
this point, excessive wear will not take place, but if this point is exceeded but 
slightly, rapid wear, or fracture of the teeth, is sure to result. For reasonable 
service, a factor of safety of 1.5 should be used if the load is uniform. Thus, 
for a forged steel gear having an elastic limit of 20,000 pounds per square inch, 

20,000 
the safe load would be = 13,330 pounds per square inch. For cast 

steel, free from apparent defects, a factor of 2 is recommended; thus, for this 



same strength in steel in a casting the safe load would be 



20,000 



= 10,000 



pounds per square inch. 

The elastic limit meant in this connection is the real elastic limit of the ma- 
terial as taken by an accurate extensometer and not by the drop of the beam, 
or by caliper measurement, as has been commercial practice. This instrument 
detects the first indications of permanent set in the test piece, showing that 



6o 



AMERICAN MACHINIST GEAR BOOK 




FIG. 51. LOCATION OF TEST PIECE. 



the safe load for that material has been exceeded ; the drop of the beam is not 
apparent for some time after this. For untreated mild steels this point is 
sometimes at one-half the drop of beam; for the higher grades, however, the 
two points are closer together. 

Such an instrument is described by T. O. Lynch in a paper on "The Use 

of the Extensometer for Commercial 
Work" read before the American Society 
for Testing Materials; Philadelphia, pub- 
lished in the Proceedings for 1908, vol- 
ume 8. 

It must be pointed out that gear steels 
should have an ample reduction of area to 
guard against sudden fracture. Test 
pieces should be cut with the center of 
tooth a little below the bottom line, say 
0.07 of the circular pitch, as illustrated by Fig. 51, as it is through this point 
that the tooth generally breaks out. 

The strength of the material in gears will be found to vary as much as 30 
per cent, in different parts of the tooth and rim, therefore a settled point for 
cutting out test pieces is necessary if uniform, safe, or accurate results are to 
be expected. It does not greatly matter if the threaded portion of the test 
piece projects into the tooth space, as it will on all gears 2^^ diametral pitch 
and finer, so long as the 0.505-inch portion of the piece is clear. When a 
0.505-inch test piece (3^^ of a square inch in area) cannot be obtained in this 
manner, make one 0.2525 inch in diameter (3^^o of ^ square inch in area), 
leaving the threaded portion % inch instead of ^^ inch, which is standard. 

Note that elastic limits given in table for wear of gear teeth is by drop of 
beam. 

THE STRENGTH OF SHROUDED GEAR TEETH 

In regard to strength of shrouded gear teeth, Wilfred Lewis submits the 
following, originally published in American Machinist of Jan. 30, 1902. 

''I do not know of any careful analyses of or experiments on the strength 
of shrouded gear teeth, but I have some recollection of a tradition in vogue 
about twenty-five years ago that from one-fourth to one-half might be added 
to the strength by shrouding. 

"There are, however, a number of cases to be considered; the shrouding 
may extend to the pitch line only or to the ends of the teeth, and it may be 
single or double. Formerly the practice of shrouding pinion was more com- 
mon than it is to-day, because the advent of steel as a cheap construction 
material makes it possible to obtain unshrouded pinions of greater strength 
than the cast-iron gears with which they engage, and now steel pinions have 
generally supplanted the old cast-iron shrouded ones which were naturally 
more roughly shaped, because harder to fit, and more difficult to assemble by 



SPEEDS AND POWERS 6 1 

reason of the shrouding. In my investigation of the strength of gear teeth I 
therefore assumed that the time for shrouded gears has passed, at least as far 
as machine tools were concerned, but they are possibly used as freely as ever 
on roll trains and some other classes of machinery, so that the problem may 
still be worthy of consideration from a practical standpoint. Rankine, in his 
'Applied Mechanics,' rather summarily disposes of the strength of gear teeth 
by assuming the load that may be carried on one corner to be all that any 
tooth is good for. When so loaded, it is shown that the corner will break off at 
an angle of 45 degrees, and the strength of a tooth of any width is then no 
greater than that of a tooth whose width is twice its hight. So, if the hight 
of a tooth is 0.65 pitch, the strength, according to Rankine, should be taken 
for a width of only 1.3 pitch. Faces wider than this would be no stronger, 
and shrouding at one end would make no difference. 

"But his assumption is untenable, because no maker of machinery who 
values his reputation will put gears into service bearing only at one end, and 
should they be so started, an even distribution of pressure is sooner or later 
effected by the natural process of wear. 

*' A comparison of strength between shrouded and unshrouded gears should 
therefore be made on the assumption of uniform distribution of pressure across 
the faces of their teeth, and for this purpose it will be expedient to neglect the 
influence of tooth forms, which would complicate and prolong the investi- 
gation, and treat all teeth simply as rectangular prisms, which may or may not 
be supported by shrouding. Rankine and Unwin have both been contented 
to estimate the actual strength of teeth as though they were rectangular 
prisms, and, although this is far from the truth, it is certainly more admissible 
as a basis of comparison for another variable than as an approximation for a 
direct result. The effect of this assumption will be to exaggerate the value of 
shrouding, and for the present it will be sufficient to indicate roughly the 
maximum benefit to be anticipated. 

''In Fig. 52 a gear tooth is shrouded at one end, and the problem is to 
determine its strength as compared with the same tooth not shrouded. For 
convenience, the thickness, or half the pitch may be taken as unity, and the 
hight as 1.3. The load W is assumed to be applied uniformly along the end 
of the tooth over the face b, making the full load bW . If there were no 
shrouding, the strength of this tooth would be measured by the transverse 
resistance at its root t, and if broken at the root as shown in Fig. 53, we may 
consider how much strength could be given to it by shrouding alone. 

"A tooth thus broken would have some strength as a cantilever imbedded 
in the shrouding, but more as a shaft subjected to torsion, and for the shape 
here assumed the torsional strength alone will probably exceed the combined 
strength due to torsion, and flexure for any actual shape. 

"The rectangular tooth whose sides are h and / cannot be treated as an 
ordinary shaft because its neutral axis is at one side instead of, as usual, at the 
center of gravity. It must therefore be treated as one half of a shaft whose 



62 



AMERICAN MACHINIST GEAR BOOK 



sides are / X 2/? or i X 2.6, for which the moment of resistance is about o. 65, 
where S is the shearing stress at the end of a tooth. For the unit load W we 
have 0.66" = i.T,W , or S = 2.17!^, and for the width b we have Si = bS = 
2.17 bW. 



— ^b- 



B 






9 


^^S 



n 



jL 



-t-* 



rn 



FIG. 52. FIG. 53. 

DIAGRAMS ILLUSTRATING THE STRENGTH OF SHROUDED GEAR 

TEETH. 

"Thus the maximum intensity of shearing Si is found to be a httle more 
than twice the full load bW. 

"On the other hand, for an unshrouded tooth the transverse stress/ at the 
root of a tooth depends only upon W and the relation is expressed by the equa- 
tion/ = 7-81^. In these terms, 

^ = 7^8' 
Si 
2.176 
whence 

_ 3-5-^1 



f 



and assuming that the shearing stress Si may be 0.8/, we have b = 2.S. This 
means that a tooth 2.8 wide has as much strength as may possibly be added by 
shrouding at one end, but the question remains to be considered, under what 
conditions and to what extent can this possible strength be made effective? 
The development of stress is always accompanied by strain, and in the case of 
a shrouded tooth the unit load W must be divided between torsion and flexure. 
Obviously, if the tooth is very long, its stiffness under torsion will be so little 
as compared with its stiffness under flexure, that the benefit from shrouding 
will be inappreciable, and on the other hand if very short, the torsional stiff- 
ness will be preponderate. When a uniform distribution of load has been 
attained, as it must be by the action of wear, that part of a tooth farthest from 
the shrouding will sustain the greatest transverse stress and the load W will be 
divided at all points along the face of the tooth between torsion and bending 
directly as the stiffness encountered in these two directions or inversely as the 
relative strains. Each strain is relieved by the other, but the limit of strength 
is reached when either attains its maximum. 

" Considering a cantilever loaded at the end, we have for the deflection y, 
under the stress/, 

y = "^' 
1.5 /r. 



SPEEDS AND POWERS 63 

where /? = 1.3 and E is the modulus of elasticity for flexure. Substituting 
this value of h we have 

y = ~^- (I) 

^'Considering the tooth as a rectangular shaft in torsion, it will be seen 
that the shearing stress for a distributed load decreases from the maximum S\ 
at the shrouding to nothing at the other end of the tooth. For the unit load 
W the shearing stress is S, for an element dx, the stres is Sdx and for this 
stress at the distance x from the shrouding, the torsional deflection 

J O Ji' (J/ %K^ 

where G is the modulus for shearing. The total deflection for the load dis- 
tributed over a length x therefore is 

or for the face h we have 

"But since G = 0.4E, we may write 

''The value of 6* has been found to be 5 = 2.17 IF, and we have also found 
= 7.8PP, therefore 

7.8 •^' 

= 0.28/. 
and 

^ ~ SE' 
may be written 

0-35/ b^ / \ 

"The deflection of an unshrouded tooth under the load W has been shown 
by equation (i), and, dividing this into equation (3), we have for the relation 

between y and z ~ = 0.36^. 

y 

For a very narrow tooth, letting b = i, we have z = o.T,y; but since ^^ and )/ are 
necessarily equal, when the tooth under consideration is attached at its root 
and also to the shrouding, the load W will be supported at both points, and it 
will necessarily be divided between them in the proportion of y to 2;, or as i to 

W 

3. The shrouding will carry — , or 0.77TF, and the root of the tooth 0.23TF. 

"Similarly making, ^ = 2, or one pitch, we have z = i.2y, and at this 



64 AMERICAN MACHINIST GEAR BOOK 

point the shrouding will carry 0.45W and the root of the tooth 0.55TF. Also, 
when & = 3, we have z = 2.73^, reducing the load on shrouding, to 0.273' 
and increasing the load at the root to 0.73!^. Kth = 4, or 2 pitch, z = 4.81/ 
the load on shrouding drops to 0.17IF and the load at the root rises to 0.83!^. 

''The average width of gear faces is probably about 2.5 pitch, and for b = 
5 we have about 0.12!^^ carried by the shrouding and 0.88TF carried by the 
tooth acting as a cantilever. We may therefore conclude that the strength of 
an ordinary pinion shrouded at one end only is not increased more than 1 2-88 
or about 13 per cent., by the shrouding. Indeed, this is only the result of a 
first approximation, and for the successive proportions of W thus credited to 
the shrouding new values of z might be estimated to be used as the basis of a 
second approximation. But we will not continue the process — it is sufficient 
to know that our valuation of the effect of shrouding is high. A double- 
shrouded pinion running with a gear whose face is 2.5/?' will be about 3 pitch 
between shroudings and its strength will be about the same as that just found 
for ^ = 3. An ordinary pinion will not therefore be increased in strength by 
double shrouding more than 37 per cent., and it is probably safe to say that a 
more elaborate investigation will reduce the additional strength to 10 per 
cent, for single shrouding and 30 per cent, for double shrouding. 

"When the shrouding extends to the pitch line only, the shearing strength 
of its attachment to a tooth is reduced, but the elastic relations upon which 
the strength at the root depends remain practically the same. In this case 
the shearing strength instead of the transverse strength limits the strength of 
a tooth, and the strength is apparently less than for full shrouding. 

"The effect of shrouding is clearly to prevent the adjacent part of a tooth 
from exercising its strength as a cantilever. The shrouding carries what the 
tooth itself might carry almost as well. A heavy link in a light chain adds 
nothing to the strength of the chain, and teeth which are not strong all over 
need not be strengthened in spots. A little more face covering the space 
occupied by shrouding is more to the purpose for durability as well as for 
strength, and when this fact is appreciated I believe the practice of shrouding 
will disappear in rolling mills, as it has done in machine shops. 

"In regard to the working stress allowable for cast iron and steel, I may 
say that 8,000 pounds was given as safe for cast-iron teeth, either cut or cast, 
and that 20,000 pounds was intended for ordinary steel whether cast or 
forged. These were the unit stresses recommended for static loads, and as the 
speed increased they were reduced by an arbitrary factor, depending upon the 
speed. 

" The iron should be of good quahty capable of sustaining about a ton on a 
test bar i inch square between supports 12 inches apart, and of course the 
steel should be solid and of good quality. The value given for steel was 
intended to include the lower grades, but when the quality is known to be 
high, correspondingly higher values may be assigned. 

"In conclusion I may say that the crude investigation here given seems to 



SPEEDS AND POWERS 65 

justify the traditions referrred to that from J^ to 3^-2 may be added to the 
strength of teeth by shrouding. If the teeth are very narrow, J^ may be 
added, but generally, I beUeve, 3^^ is enough and since writing the above I 
find that D. K. Clark almost splits the difference by adding }/^ for double 
shrouding. But the development of the full strength of gear teeth depends 
nearly as much upon the strength and stiffness of the gear journals as upon 
the teeth themselves, and no rules can be given for indiscriminate use." 

NOTES ON SHROUDED GEARS 

The use of shrouded gears, except in the case of cast units, has been dis- 
couraged, heretofore, by the difficulties imposed in making the shrouds inte- 
gral parts of the gear, tying the gear teeth securely to the rim under the teeth. 
In machining generation of gears, the formation of such shrouds cannot be 
accomplished, so that if shrouding is resorted to the shrouds have to be made 
independent rings and subsequently attached in some manner to the un- 
shrouded gear. This is a difficult and expensive task at best and it is quite 
problematic whether the slight increase in gear strength realized — the increase 
in strength being only a fraction of that attained could the shrouds be made 
integral parts of the gear — is justified. 

The recent commercial development of the rolling method of gear produc- 
tion — see Section XVI — entirely alters this situation, for in the rolling process 
shrouds are formed on the forged gear blanks. These webs, tying the forged 
teeth to the gear rim, are cut away in the production of unshrouded rolled 
gearing, but they may be retained in their entirety or in any proportion for 
the purpose of increasing the strength of the gears with rolled teeth. This 
distinctive peculiarity of rolled gearing may exert considerable influence on 
the design of gearing in the future. The addition of the shrouding not 
only increases the safe load the gears can carry but also increases substantially 
the speed at which they can be operated. Its chief advantage is, however, 
the fact that by properly proportioned shrouding the individual power capac- 
ities of the gears constituting a train may be brought up to that of the 
strongest gear — the weakest made the equal of the strongest. 

WEAR OF GEAR TEETH 

The Lewis formula is the only accurate method of figuring the power of 
gears so far as the strength of the teeth is concerned, but takes no account 
whatever of wear, and the value of the tooth surfaces to resist crushing of the 
material. Trouble from this source is a common experience, although not 
properly understood, as it is sometimes difficult to account for the 
'mysterious" failure of gears that were apparently of ample strength. It is 
noticed that gears generally fail through wear and not by fracture of the teeth, 
also that the teeth often break at a load far below that which is considered 
safe. More attention, therefore, should be given to this point. 



66 AMERICAN MACHINIST GEAR BOOK 

A certain combination of diameters will carry but a certain load per unit 
of face, irrespective of the pitch of the gears, so that there is no gain in an 
increased pitch above that just sufficient to resist fracture. This pitch may- 
be found as usual by the Lewis formula, but the actual strength of the material 
should be used. The material in a i-inch pitch tooth is stronger proportion- 
ately than the same material in a 2-inch pitch tooth. This is caused by the 
fact that nearer the exterior the material is stronger and of a closer grain, due 
to rapid cooling. A tooth is stronger at the top than at its root. It would 
seem as if tests of material should be made for flexure and not for tensile 
strength, as the tooth breaks through bending. The average ultimate 
strength of cast iron for flexure is 38,000 pounds per square inch, while the 
tensile strength is 24,000 pounds per square inch. 

Aside from this feature the surface hardness should also be considered 
irrespective of the strength of material. For instance, a pressure of 5,000 
pounds per unit of contact would be allowed for a case-hardened steel surface, 
while but 1,500 would be allowed for the same material in its unhardened 
condition. 

The relative hardness of material, in conjunction with the co-eJ0&cient of 
friction for different grades and hardness of material engaging, will supply 
the safe load A per unit of area. 

It is true that the arc of rolling contact in gears is very small; the balance is 
sliding contact, which increases proportionately over the rolling contact as the 
pitch points separate, or as the tooth disengages, and decreases as the tooth 
enters contact until the pitch points again engage where it is rolling. 

The wearing qualities of the teeth depend greatly upon their condition 
when put into service. If a little care is used to obtain a smooth surface 
at the start and allow the teeth to find their proper bearing, the gears will wear 
indefinitely longer than if put under full load when new, no matter how 
accurately the teeth are cut. Also a gear once started to cut can often be 
saved by the timely application of a fine file, finally smoothing the teeth with 
and oil stone. 

A series of experiments to determine the proper load per unit of contact 
{A ) for different grades and hardness of material used would certainly lead to 
a fuller knowledge of the capacity of gears for the transmission of power and 
leave less to supposition on the part of the designer. On account of the pecul- 
iar nature of the tooth contact it is quite likely that the best manner to reach 
accurate results would be with gears made from th ematerials under consid- 
eration. The values given in Chart 2 are the best obtainable at the present 
writing. 

The idea of limiting the load to the proportion of the gear diameters 
irrespective of the pitch (for a unit of face) may at first appear startling, but 
when we consider that the radii from which the tooth is drawn are always 
proportional to the pitch diameter of the gear and not to the pitchy and that 
the teeth in contact are actually two cylinders rolling and slipping upon each 



SPEEDS AND POWERS 



67 



Other, it appears more reasonable. See Fig. 54. It should be understood, 
however, that the diameter and position of these rollers change constantly 
throughout the contact, and that a gear made in strict accordance with Fig. 
54 would not give a uniform movement. It illustrates the principle however. 





FIG. 54. GEAR TOOTH ACTION. 



FIG. 55. CURVATURES. 



To secure safe results the flank or shortest radius is used in these formulas. 

The following is the gist of an article by Harvey D. Williams, in the 
American Machinist, with its application to gear transmissions: 

"The curvature of a plane curve is defined by mathematicians as the 
change of direction per unit of length, and is equal to the reciprocal of the 
radius of curvature at the point considered. Thus in going once around a 
circle of radius R the distance traversed is the circumference iirR, while the 
change of direction is in circular measure or radius 27r. 

"The change in direction per unit of length is therefore 



27r 



2TrR R 



"Accordingly the curvature of a 2-inch circle is i, that of a i-inch circle is 
2, that of a 3^^-inch circle is 4, and that of a straight line is — = 0, etc. . . . 

"The curvature of a straight line being zero, that of an arc may be said to 
be its curvature in relation to the straight line, or its relative curvature to the 
straight line. Similarly, in comparing the curvature of two arcs, it will be 
convenient to use the term 'relative curvature' instead of the difference of 
curvature, meaning thereby the algebraic difference of the curvature as 
dimensioned in Fig. 55. 

"It will be seen that when two plane curves are tangent to each other ex- 
ternally the relative curvature is to be found by adding the respective curva- 



68 



AMERICAN MACHINIST GEAR BOOK 



tures, and when they are tangent internally the relative curvature is to be 
found by subtracting. 

''The amount of contact between plane curved profiles is measured by the 
reciprocal of the relative curvature." See formulas 21, 22, and 23. 

In each of the cases shown by Figs. 56, 57, and 58 the contact is 4, and if 
these profiles were made of the same material and the same width of face, 
they would be equally efficient as regards their ability to withstand pressure. 




Curvature 
Contact= 



^S^^^^^^^^^^ 



00 



FIG. 56. 




Curvature =H 
Contact = 



T^ =^ 



FIG. S7- 
CONTACTS OF THE SAME CURVATURE. 




,^^ Curvat ire=H 



Coatact: 




FIG. 58. 



Let 
C 

/ 
V 
A 
D' 

d' 
a 

W 



DEVELOPMENT OF FORMULAS 

contact. 

flank radius of the gear. 

flank radius of the pinion. 

face width. 

velocity in feet per minute. 

safe crushing load per unit of contact. 

pitch diameter of the gear. 

pitch diameter of the pinion. 

angle of obliquity. 

safe load on the tooth to resist crushing and wear. 

safe load on the tooth to resist fracture. 



Then 



and 



K^ = — stji a. 

2 



1 d' • 
r^ = — stn a. 

2 



(19) 
(20) 



SPEEDS AND POWERS 69 

For spur gears 



For internal gears 



For racks 



c = 


I 






I I 








4- 






c = 


I 






I I • 








fi i^i 






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- ^/- (aoS-.) 



(21) 



(22) 



(23) 
(24) 



It is desirable that the gear and pinion should wear equally to avoid the 
necessity of engaging a new pinion with a partly worn gear, thereby decreas- 
ing the life of both. It is assumed that wear is proportional to the hardness 
of the material; obviously the pinion should be harder than the gear in 
proportion to the ratio of the drive. Therefore, to secure equal wear in a 
pair of gears having, say, a ratio of 4 to i , the pinion should be made four times 
as hard as the gear. 

I have thought that a hard pinion would tend to preserve a softer gear, but 
as no data are found to sustain this theory, and as the value calculated for 
Chart I tends toward a much softer gear than was originally thought proper 
to make the best wearing combination for certain ratios, this has not been 
taken into account. It is assumed, therefore, that a hard pinion will neither 
preserve nor influence the wear of a softer gear. Therefore, it may be 
assumed safely that a hard gear will not influence the wear of a softer pinion. 
Chart I is made on this basis. The wear of gear and pinion is determined 
independently if the proper combinations of hardness are not used. 

The wear is based entirely on the pinion hardness, the gear performing the 
same amount of work, but having less wear on account of the greater number 
of teeth in use in proportion to the ratio of the gears. For instance, in a gear 
drive having a ratio of 4 to i, the gear may be 75 per cent, softer than the 
pinion for equal wear. 



70 



AMERICAN MACHINIST GEAR BOOK 



Thus the wear of the gear is found according to the pinion hardness that is 
proper for the ratio of the gears irrespective of the material actually used for 
the pinion. 



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Hardness of the Pinion 

For example: If a pinion of a hardness represented by 0.15 (see Chart i) 
engages a gear of 0.35 hardness, the ratio being 4 to i, the wear of the gear will 
be in accordance with the pinion hardness found opposite the line of ratio and 
over the gear hardness (0.35), which in this case is 0.17 J^. In this event the 
gear will wear out first. On the other hand, if the pinion had been of 0.20 
hardness, the pinion would wear out first. The value for the wear of the gear 
would remain 0.173^. 



SPEEDS AND POWERS 



71 



It is thought that the elastic limit of a material follows the hardness, there- 
fore the wear may be determined from the elastic limit. The points of hard- 
ness, however, are used in the accompanying charts for convenience ; Chart 2 





— 






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gives the corresponding values. The hardness values given in these tables 
were obtained' by pressing a J^-inch hardened steel ball into the surface of the 
material with a pressure of 10,000 pounds. The dimensions given are 
diameters of the indentations thus made. See Fig. 59. The comparisons 



72 



AMERICAN MACHINIST GEAR BOOK 



in hardness were made inversely to the square of these diameters. The 
elastic limit was determined for one of these values; the comparison for 
others may be found by the same inverse proportion. 





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Pitch Diameter of the Pinion in Inches 



Thus, the square of 0.20 = 0.04 and the square of 0.30 = 0.09. There- 
fore, 0.20 would be — — — 2\'i times harder than o.^o. This is a proper 
' 0.04 

combination for a gear ratio of 2}^ to i. The elastic limit for 0.20 is 60,000 

6o,ooo 
pounds per square inch. The elastic limit of 0.30 = — jy- = 26,700 

2/4 

pounds per square inch. 

For comparison with the Brinell scale; hardness value 0.22 inch in Chart 2 



SPEEDS AND POWERS 



73 



measures 4.6 millimeters; the hardness numeral for this impression being 167, 
the impression is made with a lo-millimeter ball at a pressure of 3,000 kilo- 
grams. 

As the elastic limit of cast iron is very close to its ultimate tensile strength 
the ultimate strength may be used to determine the hardness. This was at 
first very confusing before it was found that the hardness followed the elastic 
limit, as cast iron under the ball test referred to would show a hardness equal 
to machine steel of twice its ultimate tensile strength. 

I -H -I 



•/////////////////. 



^//////////yy///y 



FIG. 59. CHORD MEASUREMENT OF HARDNESS TEST. 

The values of A according to the hardness or elastic limit of the material 
(Chart 2) have been assumed as correct for gears operating 10 hours per day 
for a period of two years. If found in error their multiplier given in Table 
15 for the time and conditions of service may be shifted without changing the 
original values of Charts i and 2. This table may be elaborated to any de- 
sired extent to cover various conditions. It is evident that a pair of gears 
will not last as long fastened to the ceiling or to insecure timbers as if mounted 
upon a proper concrete foundation. There are all manner of machine con- 
structions to be considered as well as unknown overloads, the influence of fly- 
wheels and other things that are usually neglected. All this, however, is 
simple indeed when the Stygian darkness in which we are now wandering is 
considered. 



TIME or SERVICE 



For 3 months 
For 6 months 
For 9 months 
For I year. . . 
For 2 years . . 
For 3 years . . 
For 4 years . . 
For 5 years . . 
For 6 years . . 
For 7 years. . 
For 8 years . . 
For 9 years . . 
For 10 3^ears . 



UNIFORM LOAD 



CONTINUOUS 



3.00 
1.50 
1. 00 

0.75 
0.38 
0.25 
0.19 
0.15 
0.13 
O.II 
O.IO 

0.09 

0.08 



10 HOURS DAILY 



8.00 
4.00 
2.67 
2.00 
1. 00 
0.67 
0.50 
0.40 

0-33 
0.29 
0.25 
0.22 
0.20 



5 HOURS DAILY 



E5.00 
9.00 
6.00 

4-50 
2.25 
1.50 

I-I3 
0.90 

0.7.^ 
0.65 

0.56 

0.50 
0.45 



Table 13 — Multipliers for Factor 'A' 
According to the Cojiditions of Service and Desired Life of Gears 



74 AMERICAN MACHINIST GEAR BOOK 

For gears subjected to 25 per cent, overload, multiply result by 0.80; for 
gears subjected to 50 per cent, overload, multiply result by 0.70; for gears 
subjected to 75 per cent, overload, multiply result by 0.60; for gears subjected 
to 100 per cent, overload, multiply result by 0.50; for gears operating in dust- 
proof oil case, multiply result by 1.50. 

EXAMPLES 

What is the safe load for a pair of spur gears properly mounted on concrete 
foundations to operate continuously for a period of five years before replacing? 
The gears are to run in oil in a dust-proof case driving an electric generator 
making 300 revolutions per minute from a turbine revolving at 1,200 revolu- 
tions per minute. Theoverloadat no time will exceed 25 per cent. The gear 
has 84 teeth, 3 diametral pitch, 12-inch face and 28-inch pitch diameter. 
The pinion has 21 teeth, 3 diametral pitch, 12-inch face and 7-inch pitch 
diameter. The ratio is 4 to i. The circumferential speed at the pitch line 
is 2,200 feet per minute. The pinion should be four times as hard as the gear 
for equal wear and, according to Chart i, if a cast-steel gear of 30,500 pounds' 
elastic limit which is assumed to have a hardness value of 0.28, the pinion 
hardness should be represented by 0.14, which represents an elastic limit of 
120,000 pounds per square inch. This may be obtained by hardening 
chrome nickel or other high-grade steel. High-carbon steel should be 
avoided for this purpose on account of its tendency to crystallize. 

Referring to Chart 2 it is found that the temporary value of A is 7,500 
pounds. The multiplier for time of service is o. 1 5 ; the multiplier for overload 
is 0.80; the multiplier for the oil case is 1.50. Thus the final value of ^ = 
7,500 X 0.15 X 0.80 X 1.50 = 1,350 pounds. 

In the formula 

600 



W = CfA 



6.00 + V 



where 



then 



or 



W^ = the safe load in pounds (to be determined), 
C = 0.70 (from Chart 3), 
/ = the face, 12 inches, 

A = I, s so and ~ , (from Chart 4) = 0.21, 

'^^ 600+2,200^ 

W^ = 0.70 X 12 X 1,350 X 0.21 = 2,380 pounds, 

2,380 X 2,200 



33,000 



= 158 horse-power. 



The strength of the teeth must now be checked by the Lewis formula to 
guard against fracture at this load. We find by this method a safe working 
load of 317 horse-power. 



SPEEDS AND POWERS 



75 



This illustrates that the teeth are capable of carrying 317 horse-power, 
but as shown by the value W*' they would wear out in about one-half the 
specified time if such a load were applied. If the example had read " ten hours 



0.10 






























-^ 




































r 






^ 




























































































0.20 




























































— 




























































, 


— ■ 




— ' 


































































--' 


—■ 


-' 


■^ 


















































•3 0.40 

> 

00.50 

g 
























^ 














































































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3 
















y 


y 















































































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' 










/ 

















































































/ 










































































m 

§ 0.70 

^ 0.80 






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1 














































































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1 
















































































1..00 



















































































200 400 600 800 1000 1200 1400 1600 1800 2000 2200 2400 2600 2800 3000 3200 3400 3600 3800 4000 

Velocity in Feet per Minute 



CHART 4. RELATIONS BETWEEN THE QUANTITY 



600 



600 -\- velocity 



-- AND THE VELOCITY. 



per day" instead of ''continuous," all other conditions remaining the same, 
we would have : 

A = 7,500 X 0.40 X 0.80 X 1.50 = 3,600 pounds, 

3,600 X 2,200 , 

= 240 horse-power. 

33,000 

For this load, however, the teeth would be liable to fracture. 



RELATIVE IMPORTANCE OF STRENGTH AND HARDNESS 

It would appear that the actual strength of the tooth is to be a secondary 
consideration, figured only as a preventive against fracture. The real points 
to be considered are: First, the proper proportion of the gear diameters; 
second, the hardness of the material and the best combination of hardness for 
wear. George B. Grant was very near the truth in saying, "It does not 
proportionately increase the strength of a tooth to double its pitch." With 
herringbone gears it would seem that the strength need hardly be considered, 
as it is practically impossible to break out a single tooth of sufficient angle, 
an entire section must be removed. They must be worn out. 

Aside from the hardness, the value of the material to avoid crystallization 
must be considered, as gears in which the teeth are apparently extremely 
tough will often become brittle and drop off after a comparatively short 
service from this cause. For this reason high-carbon steel should be avoided 
and hardness obtained by case-hardening, or by the addition of manganese, 
nickel, chromium, vanadium, or some other hardening ingredient. 



76 



AMERICAN MACHINIST GEAR BOOK 



It is hardly necessary to add that proper lubrication adds greatly to the 
efficiency of a gear drive, except, of course, where they are exposed to brick or 
cement dust, where it is often advisable to run them dry, as the oil will hold 
particles of grit and cause the teeth to cut. A jet of air appplied at the point of 
contact is also found beneficial in such cases, as it will remove particles of grit 
from the teeth before they enter contact. 

The constructions of housings upon which the gears are mounted is of the 
utmost importance, as the absence of vibration is essential to high efficiency. 
Where the housings are insecure it is often found that a rawhide pinion will 
sometimes give better service than one made of iron, as the rawhide will give 
and absorb vibration that would destroy the harder material. 

Another point that naturally suggests itself is the proper value of a suit- 
able lubricant. It is evident that the efficiency of 95 per cent, running dry 

and 98 per cent, when immersed in oil 
does represent the total saving. Lu- 
brication means in many cases the 
difference between a successful drive 
and a failure which is not apparent 
from any superficial tests made for 
power efficiency only. 





FIG. 60. INTERNAL GEAR MESHING WITH 
SPUR. 



FIG. 61. 



LIMITING WEAR OF GEAR 
TEETH. 



The question is often asked, ''What is the life of a gear?" It is evident 
that continual service will wear out any material no matter how hard. We 
will say for example that a load of 900 pounds per unit of area will wear out a 
pair of cast-iron gears in one year's continual service. What pressure must 
be applied, therefore, to wear out these gears in six months or to allow them 
to run for five years? It would appear that this could be taken care of nicely 
by factor A, and as in Table 15, provided, of course, that other conditions 
are correct and that we have the proper analysis of our materials. We 
should determine not only what grades of material will wear best together, 
but also how long they will wear. 

A gear may be said to be worn out when the teeth have been reduced to 
not less than one -half their original thickness — this will subject them to 
the limit of ultimate stress of the material if allowed their full load according 
to the Lewis formula. If this is exceeded the teeth would be liable to fracture. 
See Fig. 61. 

It is evident that during the latter part of the life of a gear the teeth will 



SPEEDS AND POWERS 77 

wear more rapidly, as the backlash will allow the teeth to hammer with 
variations in the load or in reversing. 

It is well known that all materials are subject to what is known as fatigue, 
that is, a piece of steel that will stand an intermittent strain of 2,000 pounds 
successfully is liable to fail if this load is permanently applied. This should 
enter into out problem, as it is often required to design a pair of gears to 
transmit a certain load continuously, with the guarantee that they will render 
successful service for, say, two years before renewing. The correct solution 
of this problem would require a thorough knowledge on all the points brought 
up in this paper, also a proper determination of their efficiency, unless of 
course, the gears were made amply large, or there had been some precedent 
upon which to base the calculations. 

Another point to be considered is what difference (if any) should be made 
in the comparative hardness in favor of the pinion, so that it may have wear 
proportional to the ratio of the pair. 

IMPORTANCE OF PROPER DESIGN 

Aside from all these conditions, to obtain anything like accurate results the 
gears must be mounted in such a manner as to obviate practically all vibra- 
tion. The teeth must be accurately formed and spaced to insure that the 
impulse received by the driven gear is uniform and without variation, and the 
thickness of the teeth must be such as to avoid practically all backlash, 
except just enough to secure free operation, as teeth are often broken by 
crowding on close centers. The gears must also be properly designed to 
withstand any strains to which the teeth are subjected. The proper distri- 
bution of the material will add greatly to the wear and strength. It is well 
to have the gear as rigid as possible. It should be remembered that accuracy 
in cutting the teeth will be of little avail if they are not correctly mounted 
upon their shafts. If the shaft is a little under size, the key will cause it to 
run out of true. This will also apply when the shaft is a neat fit and the 
taper key is driven too tightly. 

In the absence of practically all experimental data I have not attempted 
to put forward anything more than a general outline of the situation and to 
bring up essential points for consideration. It is trusted that they will be 
received as such. In view of the growing importance of gears for the trans- 
mission of power the points referred to are certainly worthy of attention. 

RELATIVE STRENGTH AND HARDNESS OF HOT ROLLED GEARING 

The advent of rolled gearing, with its distinctive strength and wear 
resisting characteristics, has quite materially increased the limits placed upon 
these important properties for cut gearing and gives promise of greatly 
increasing the life and modifying the design of all types of commercial gearing. 

Tests of similar machine cut and hot rolled gears (6-inch P. D. 6-8 pitch) 



78 



AMERICAN MACHINIST GEAR BOOK 



conducted at the James Herron Laboratories, Cleveland, Ohio, for normal 
and case-hardened conditions of gears gave the following average results per 
inch face of gear — the steel of which the gears were made containing 0.20 
per cent, carbon and 0.57 per cent, manganese. 



MACHINE CUT 
GEARS 



HOT ROLLED 
GEARS 



Normal Condition 

Yield point 

Ultimate strength. . 

Hardness 

Case-hardened Condition 

Yield point 

Ultimate strength. . 

Hardness 

Penetration of case 



6,470 lbs. 

12,250 lbs. 

22 

13,529 lbs. 
17,250 lbs. 

0.02 7-in. 



7,918 lbs. 

13,645 lbs. 

26 

14,750 lbs. 
19,130 lbs. 

85 
0.03 5-m. 



In the normal condition — that is, not case-hardened — the hot rolled gears 
showed a superiority of 22.4 per cent, in yield value, 11.9 per cent, in ultimate 
strength and 18.2 per cent, in hardness; while in the case-hardened condition 
the hot rolled gear averaged higher by 9.0 per cent, in yield value, 10.9 per 
cent, in ultimate strength, 29.6 per cent, in penetration of case and very 
nearly the same in degree of hardness. The hot rolled gears were also very 
much the tougher, indicating exceptionally high wear resisting qualities. 

Photomicrographs of annealed cut gear tooth sections indicated a coarse 
structure characteristic of drop forgings with the pearlite gathered together 
in relatively large islands indiscriminately scattered throughout the metal; 
while similar photographs of hot rolled gear tooth sections showed a very 
much more uniform structure with the islands of pearlite small, well broken 
up and uniformly distributed throughout the metal. In case-hardened 
specimens, those of cut gears showed more or less free cementite on the out- 
side of the case, but in the specimens of hot rolled gears there was much less 
free cementite and a more uniform and deeper case penetration, giving the 
hot rolled gear teeth a very tough and high wear resisting surface. 

These laboratory tests indicate that the hot rolling process adds from 20 
to 25 per cent, to the strength of the gears, increases their hardness 18 or 20 
per cent, and materially betters their wear resisting qualities. These records 
have been duplicated or bettered repeatedly under normal and exacting 
working conditions. 

Photographs of magnified cross-sections of machine cut and hot rolled 
gear teeth show plainly that the 10 to 20 ton pressure to which the 
heated gear blanks are subjected in the process of rolling the teeth gives to the 
forged teeth not only a much smoother profile surface, but brings about a 
marked rearrangement of the metal structure, producing a dense almost 
fiber-like arrangment in a trussed formation about the periphery of the gear 



SPEEDS AND POWERS 



79 



which serves to tie the forged teeth to the body of the gear and to equalize 
any warpage effects in subsequent heat treatment, so ehminating the need of 
heavy clamping devices for holding the gear true when quenching. 

SPEED OF SPUR GEARS 

When the question is asked, ''What is the greatest circumferential speed 
at which a spur gear may operate?" we are told, "When 1,200 feet per minute 
is exceeded either rawhide or herringbone gears must be used, and even when 
properly mounted 3,000 feet is the limit of speed for any gear." 

To illustrate the fallacy of this statement consider a pair of spur pinions, 
each of 12 teeth, 6 pitch, 2-inch diameter, running at a speed of 1,200 feet per 
minute. A moment's consideration will 
show that the noise generated by such 
a drive would be excessive, as this would 
mean 2,280 revolutions per minute. 
These gears would make their presence 
known at a speed of 400 feet per min- 
ute, which represents 761 revolutions per 
minute. 

At first thought it appears that the 
number of teeth in contact would repre- 
sent the comparative speed value, but, as 
it is sometimes possible to obtain as many 
teeth in contact in a small pair of gears as 
in a large pair, owing to the difference in 
pitch, the proportion of gear diameter fig. 62. diagram illustrating method 

1 , , . . , OF DETERMINING NUMBER OF TEETH 

must also be taken mto account. As the ^^ contact. 
proportionate value of the gear diameter 

is represented by the number of teeth in contact, the pitch remaining con- 
stant; this value may be gaged according to the circular pitch. Fig. 62 will 
illustrate this. 

The relative speed value is found in the product of the number of teeth in 
contact and the circular pitch, or np\ 

n = number of teeth in contact, 
p^ = circular pitch. 

The next step is the determination of a factor from actual practice, p, 
which multiplied by the product of the pitch, p', and the number of teeth in 
contact, n, will give the safe speed. The formula now becomes: 

Safe speed = p'np. (26) 

The factor p may be made to cover almost any type of gear or condition of 
service. If the safe speed is known for certain machine construction, the 
value of the factor p may be determined as below for similar drives. 

Safe speed 




Value of p = 



p'n 



(27) 



8o 



AMERICAN MACHINIST GEAR BOOK 



STYLE OF GEAR 



Spur gear, pattern molded. 

Spur gear, machine molded 

Spur gear, commercial cut 

Spur gears, cut with exact cutters, accurately spaced 

Spur gears, cut stepped teeth 

Spur gears, fiber 

Spur gears, rawhide 

Herringbone gears, angle of spiral, lo degrees 

Herringbone gears, angle of spiral, 20 degrees 

Herringbone gears, angle of spiral, 30 degrees 

Herringbone gears, angle of spiral, 45 degrees 



COMMERCIAL 


GENERATED 


CUT GEARS 


TEETH 


to 300 




1 10 to 450 




600 




700 


800 


820 




900 


1000 


1000 




700 


1 100 


800 


1400 


1 100 


1900 


2400 


400c 



Table 14 — Values of Factor p 



NUMBER OF TEETH IN CONTACT 



As the necessary formula to determine the number of teeth in contact is 
considered too cumbersome for practical use, a graphical solution is given, as 
follows: 

Referring to Fig. 62, the length of contact is measured between the inter- 
section points of the line of pressure and the addendum circles of the gear and 
pinion, or between a and b. In case this intersection falls outside the inter- 
section of a line drawn at right angles with the pressure line to the center of 
the gear as at the point c, the contact is measured from the point c, as this 
indicates that the distance a c must be deducted for interference. The gear 
tooth is rounded from this point out, or the flank of the pinion tooth is under- 
cut to accomplish the same purpose. 

If, on the other hand, this point falls outside the intersection of the 
pressure line and the addendum circle as at c' this extra length must be 
deducted, as there is no contact until the point of the mating tooth has passed 
the addendum circle. In order that the number of teeth in contact can be 
stepped off, the lines o^c and ob are extended to the pitch or to points e and/. 
The length of contact is then measured from e to c? on the gear, and from d to/ 
on the pinion along this pitch circumference. This distance divided by the 
circular pitch equals the number of teeth in contact. 



LIMITING SPEEDS 

Estimate the maximum speed to avoid danger of fracture for the best type of 
gear and condition of service as 500 feet per minute per 1000 pounds safe 
working stress of the material of which the gear is constructed. Thus the 
maximum speeds would be as follows: 

4,000 feet per minute for cast iron of 8,000 pounds per square inch. 
8,000 feet per minute for cast steel of 16,000 pounds per square inch. 
10,000 feet per minute for machinery steel of 25,000 pounds per square inch. 



SPEEDS AND POWERS 8i 

To attain anything like these speeds, however, the gears must be excep- 
tionally accurate and well balanced, also the housings must be sufficiently 
heavy to obviate practically all vibration. The restriction placed by the 
above limits, however, will avoid the possibility of allowing the higher speeds. 
According to a series of experiments made by Prof. Charles H. Benjamin, 
American Machinist, December 28, 1901, page 1421, the bursting speed of a 
solid, cast-iron gear blank is found to be 24,000 feet per minute. The 
centrifugal tension at this speed is 15,600 pounds per square inch. The 
same wheel, split between the arms, burst at an average speed of 11,500 feet 
per minute. 

SPEEDS AND POWERS 

The superiority of hot rolled gears in the matter of strength, toughness and 
uniformity of metal structure permits the attainment of somewhat higher 
speeds than are safe for cut gears, particularly as they are of relatively small 
diameter and are, consequently, proportionally of more rugged construction.* 
The modification in metal structure brought about by the heavy pressure 
exerted on the semi-plastic metal not only serves to tie the forged teeth to the 
gear rim but also to form a confining ring of high tension metal about the 
gear body, increasing the resistance of the gear to the action of centrifugal 
force. Exactly what is the permissible allowable increase in limiting speed 
has not as yet been determined, but in the case of unshrouded gears it can be 
safely taken as 20 per cent. In the case of shrouded rolled gears this increase 
can be exceeded without danger, the additional increase depending largely 
upon the height of shroud and probably being limited to another 30 per cent. 
— a total increase of approximately 50 per cent. 

CONSTRUCTION OF THE GEAR 

Approximate multipliers should be used for various designs in reference 
to limiting speeds. 

Hot rolled gears (shrouded) up to 1.50 

Hot rolled gears (unshrouded) 1.20 

Properly proportioned solid gear i.oo 

Gear split through the arms 0.75 

Gear split between the arms 050 

Link flywheel construction 0.60 

With the exception of the values given for hot rolled gearing, these 
values correspond closely to those given by the Fidelity and Casualty 
Company, which sets the limit of speed for a solid flywheel (cast iron) at 
6,000 feet per minute. No value is given for the wheel split through the arms. 

The principal cause of failure (as pointed out by Professor Benjamin in the 

* Reginald Trautschold. 
6 



82 AMERICAN MACHINIST GEAR BOOK 

article above referred to) in gears split between the arms is the necessity of 
placing bolting lugs on the inside of the rim. These lugs naturally tend to 
increase the stress at their point of location and fracture the rim in this 
locality at correspondingly low speeds. 

LOCATION OF THE GEAR 

The influence of the location on the speed cannot be well determined. In 
general, however, for gears mounted upon insecure foundations or secured to 
the wall or ceiling of light buildings more or less allowance must be made. 
One case in mind is a lo-horse-power motor direct-connected to a Une shaft 
making 150 revolutions per minute. The motor is securely bolted to the 
joists: The gears have 106 and 22 teeth, respectively, 4 diametral pitch, 5-inch 
face, cast iron and rawhide; the speed is 1,050 feet per minute. These gears 
were exceptionally noisy and were replaced by herringbone gears of the same 
normal pitch, with the angle of spirality 20 degrees. The gear in this instance 
was cast iron and the pinion machinery steel. These gears were no better 
than the first pair and were replaced by gears with a spiral angle of 45 degrees. 
These proved to be but little better, and as a last resort a pair was installed 
with an angle of 76 degrees. These gears were fairly quiet when operating 
under full load, but very distressing when running light. This condition is 
sometimes found in spur gears with excessive backlash. It was noticed that 
the teeth gave evidence of rapid wear due to the reduced face contact. 
Rubber cushions placed under the motor and shaft hangers and filling the 
space between the hub and rim of gear with wood made no perceptible dif- 
ference in the noise of this drive, which was finally abolished. 

In this connection I have knowledge of a pair of herringbone gears, cast 
iron and machine steel, 80 and 24 teeth, 6 normal pitch, 20- and 6-inch pitch 
diameters, angle of spiral 45 degrees that are practically noiseless at a speed of 
3,150 feet per minute. At this speed, however, the load is comparatively light, 
These gears are driven by a lo-horse-power motor and are entirely 
satisfactory. 

RELATION OF PRESSURE TO SPEED 

The question naturally arises, "In what way does the tooth pressure 
influence the speed of gears?" From a power standpoint this is ordinarily 

taken care of by Mr. Earth's expression 7 X~F ^^^ ^^^^ ^^^^ ^^^ ^^ ^^^ 

speed at which noise may be avoided, which is the subject of this paper. 

It has been assumed that the tooth pressure allowable at the speeds given 
by the formula (26) are within the limits placed by the foregoing formula (24) 
for wear. 

As previously pointed out " the strength of teeth in herringbone gears need 
hardly be considered, as it is impossible to break out a single tooth provided 
the angle be great enough to engage another tooth before the first lets go." 



SPEEDS AND POWERS 83 

However, the actual contact, which depends upon the angle, should be used 
instead of the actual face when determining the load for wear, as increasing 
the angle decreases the noise, but increases the wear. 

ILLUSTRATIVE EXAMPLE 

Required the safe speed of a pair of solid steel spur gears of the 
following dimensions: 

Gear, 80 teeth, 2-inch pitch, 6-inch face, 50.93-inch pitch diameter, pin- 
ion 48 teeth, 2-inch pitch, 6-inch face, 30.558-inch pitch diameter. 

Cutters for these gears to be made for the exact number of teeth. 

The computations for speed involve the number of teeth in contact = 
2.5, value of p according to Table 16 = 800, safe speed = p'np = 2 X 23>^ X 
800 = 4,000 feet per minute. According to factors given for limiting speeds, 
these gears would be amply safe to resist fracture, but would be at the extreme 
limit for cast iron of a safe stress of 8,000 pounds per square inch. The com- 
putations for wear involve, assuming that the pinion is made from steel of 
an elastic limit of 50,000 pounds per square inch, the limiting tooth pressure 
at this speed would be equal to the value PF^ as follows: 

W'^ = CfA 7 -— =^ = 2.4 X 6 X 3,000 X 0.13 = 5,600 pounds, 

in which 

C = 2.4, 
/=6, 
A = 3,000, 
and 

600 

The computations for strength involve the use of the Lewis formula, based 
on carrying the entire load on one tooth, 

W = Sp'fy = 25,000 X2X6X0111 X 0.13 = 4,329 pounds, 

in which 

S = 25,000 pounds per square inch (one-half the elastic limit), 
p^ = 2 inches, 
/ =6 inches, 
and 

y =0.111 for 48 teeth. 

As the load should not exceed the strength of the tooth, this pressure is the 
limit to be used. The corresponding horse-power transmitted would equal 

WV _ 4,329 X 4,000 



33,000 33,000 



= 524 horse-power 



84 AMERICAN MACHINIST GEAR BOOK 

These gears if properly mounted should be satisfactory at a speed of 4,000 
feet per minute, the value TF^ indicating that they could be used for 10 hours 
per day service for two and one-quarter years at a uniform pressure of 4,600 
pounds. If longer life or service per day is required the load on the teeth 
must be reduced accordingly. 

HIGH SPEED GEARING* 

For the transmission of power it frequently becomes necessary to use 
toothed gearing, subjected to high peripheral speed conjointly with high 
pressure per unit of tooth contact, and the object of these remarks is to 
record what has been successfully done in recent years, as much higher speeds 
are now successfully attained than formerly. Considered in a static sense, 
the gear tooth satisfies the condition of stress if it is proportioned to endure 
forces acting transversely on it, and the pressure per unit of contract is not 
of such intensity as permanently to deform the curved bearing surface of the 
teeth. When in motion, the curved surfaces slide upon each other as they 
enter and leave contact, and when this sliding action is accompanied with 
high pressure, the limit of endurance is soon reached, and in the case of the 
inferior materials this occurs at comparatively low speeds and pressures. 
In addition to this, more or less impact usually occurs, especially when the 
resistance is of a fluctuating character or the loads are suddenly applied. 
The effects of this hammering action are discernible by a flattening of the 
curved faces of the teeth, after which the proper engagement of the teeth 
ceases and the gear is speedily destroyed. 

To prevent this, it is desirable to cut the teeth so accurately that no side 
clearance or '' backlash" exists, and this is now usually done on first-class 
gearing of even the largest dimensions. Owing to the low elastic limit of cast 
iron and the bronzes we cannot expect these metals to endure so high a pres- 
sure as steel, and steel appears to be the most trustworthy material to endure 
the highest pressures and speeds. This assertion, however, does not apply to 
all grades of steel. Soft steel surfaces abrade or cut very readily despite all 
methods of lubrication, and surfaces of this material should never be allowed 
to engage in sliding contact. Gearing of soft steel is usually destroyed by 
abrasion at quite moderate speeds. Rolling-mill pinions of steel, containing 
0.3 per cent, carbon, have been destroyed in a few months, whereas the same 
pattern in steel of 0.6 per cent, carbon has done similar work for several years 
without distress. Of course it is necessary to shape the teeth to a proper 
curve to insure proper engagement and uniform angular velocity. 

Some years ago there was required suitable gearing to connect the engines 
to a rolling mill in this vicinity. The diameters of the wheels were 37.6 and 
56.4 inches respectively. They were intended to revolve at speeds of 150 
and 100 revolutions per minute and expected to transmit about 2,500 horse- 

* A paper read before the Engineers' Club of Philadelphia, by James Christie. 



. SPEEDS AND POWERS §5 

power The character of the service was such that renewal was a serious 
matter and long endurance very desirable. A high grade of steel was selected 
especially in the pinion, in which the greatest wear would occur, and which, 
owing to the location, was the most difficult to replace. The pinion was 
forged from fluid compressed steel of the following composition: 

Per cent. 

Carbon 

Manganese ^ 

Silicon ' 

Phosphorus and sulphur, both below o • ^3 

The spur wheel was an annealed steel casting: ^^^ ^^^^^ 

Carbon °-47 

,^ o.oo 

Manganese 

Phosphorus and sulphur, both o • °5 

The tooth dimensions were: Pitch, 4.92 inches; face, 24 inches. See Fig. 

' These were accurately cut with involute curves generated by a rolling 
tangent of 16 degrees obUquity. No side clearance was allowed. After 
starting the mill, it was found that a higher speed was practicable than was 
originally contemplated. Higher pressures on the teeth were also applied, 
so that ultimately about 3,300 horse-power was transmitted through the gear- 
ing corresponding to a pressure of nearly 2,100 pounds per inch of face. The 
speed was variable, but occasionally attained a velocity of 260 revolutions per 
minute for the pinion, corresponding to a peripheral velocity of 2,500 feet per 
minute. This gearing has been in constant operation for several years and 

behaves satisfactorily. ^ j -u a 

The highest recorded speed for gearing that I can recall is that described 
by Mr. Geyelin in the Club ''Proceedings" of June, 1894. The mortise 
bevels had a peripheral velocity of 3,900 feet per minute, but the pressure 
per inch of face was only about 680 pounds, the diameter and speed bemg 
made high to reduce the pressure on the teeth. I understand that the life- 
time of these bevels is not long. If made of a grade of steel, as previously 
described, their diameter and speed could be considerably reduced and pro- 
longed endurance would be realized. 

About the same time No. 63 was installed a similar appUcation was made 
to another mill, the gear having a different speed ratio, and the angular 
velocity being lower. See Fig. 64. p.^.^^ ^^^^^ 

percent. percent. 

Carbon °-9° °-^° 

T»/r O 64. 0.04 

Manganese ^ 

A much larger set had been previously employed, transmitting about 2,400 
horse-power at 750 feet per minute peripheral speed, involving a pressure 



86 



AMERICAN MACHINIST GEAR BOOK 




Fig. 65 | 

T 

EXAMPLES OF HIGH SPEED TOOTH GEARING. 

per inch of face of 3,500 pounds. This latter pair were 4 feet and 8 feet 

respectively, 73^^-inch pitch, 30-inch face, cut with involute teeth of 14 
degrees obliquity. See Fig. 65. 

Pinion Gear 

Carbon 0.52 0.42 

Manganese 0.55 0.73 

Silicon 0.107 0.279 

Phosphorus 0.022 0.078 

Sulphur 0.02 0.05 



SPEEDS AND POWERS 87 

These gears have all rendered excellent service, and to-day are apparently 
as good as at the beginning. 

As considerable expense is involved in cutting large gears of hard steel, it 
is sometimes practicable to rough-cut the gear after it is made as soft as pos- 
sible by slow cooling, a higher degree of hardening being imparted before final 
finishing by air hardening or rapid cooling from the refining heat. This is 
not infrequently done in the case of screws and gears of moderate dimensions. 
In this event it is desirable to have the ratio of manganese low — say, not 
over 0.5 or 0.6 per cent. — as a high manganese content seems to impart a 
permanent hardness that is not reduced by slow cooling. 

It appears to be practicable to maintain sliding surfaces of steel if one of 
the surfaces is hard, even if the other is comparatively soft, but for steel 
gearing for ordinary purposes I would suggest the use of steel not less than 0.4 
carbon. If the speeds and pressures are unusually high, a much harder grade 
of steel becomes necessary. When a small pinion engages with a large wheel, 
the former alone can be made of high grade steel approaching to a carbon 
content of i per cent. When extreme speeds and pressures become necessary, 
the best results will be found by using in both wheels steel having a carbon 
content approaching i per cent., or an equal hardness, obtained by lower 
carbon and high manganese or other desirable hardening addition. With 
gearing accurately cut from steel of this character and securely mounted, it 
is believed that reasonable endurance will be obtained when the product of 
speed and pressure, divided by pitch, each within certain limits, does not 
exceed 1,000,000: For example, a speed of 3,000 feet per minute and 1,600 
pounds per inch of face, or vice versa for gear of 5-inch pitch, assuming, so 
far as we know, a maximum speed of 5,000 feet per minute for gear of any 
pitch, and permissible pressure to be proportional to the pitch. 

This statement that speeds and pressures are reciprocal, or as one is 
increased the other must be reduced, in a fixed ratio, may not strictly be a 
rational one, but in a broad and general sense it is correct within the usual 
limits of practice. 

It will be understood that such a generalization as herein stated would 
apply to pinions having a liberal and not the minimum number of teeth. 

In the discussion of Mr. Christie's paper, Mr. E. Graves gave particulars 
of three duplicate sets of cast-steel bevel wheels. The pinions are the drivers 
and are 57.39-inch pitch diameter and have 36 teeth, 5-inch pitch, 20-inch 
face. The wheels are 74.8 inches diameter and have 47 teeth. The teeth 
are carefully cut to involute lay-out and are 3.43 inches high. The normal 
speed of the pinion is 360 revolutions per minute, giving 200 revolutions per 
minute to the wheel and nearly 4,000 feet circumferential speed on the pitch 
line. The horse-power transmitted is 1,300. Assuming the entire load to 
be distributed along the outer end of one tooth, the fiber strain would be 
2,100 pounds per square inch at the root of the tooth. 

The pinion is mounted on the upper end of a lo-inch shaft, 148 feet long, 



88 AMERICAN MACHINIST GEAR BOOK 

with a turbine wheel at the lower end. Both shafts extend through the gears 
and are supported in a massive bridge casting with adjustable bearings. The 
gears are enclosed in a casing and are lubricated with oil fed under pressure 
through several jets applied just in front of the teeth as they mesh together. 
The gears have been in service for five years, but have not been entirely 
satisfactory. Their wearing power in the sense of resisting abrasion is satis- 
factory, but the teeth break. This breakage is confined to the pinion, the 
nature of the break being the same in all cases, beginning at the large end, 
cracking around the root and following along the tooth. The quality of the 
steel in castings is the ordinary commercial article. The widest variation 
in analysis observed is, in one instance: 

Silicon 0.25 

Sulphur 0.036 

Phosphorus 0.071 

Manganese 0.74 

Carbon 0.3 1 

Another : 

Silicon 0.27 

Sulphur 0.03 

Phosphorus 0.032 

Manganese 0.80 

Carbon o. 23 

As is to be expected, the softer metal has resisted breaking the longer. In 
two sets of these gears the resisting work is of a varying nature with sudden 
and wide fluctuations; in the third instance the working is more constant. 
This variation of conditions does not seem to have influenced failure, as the 
teeth have broken in all the sets. 

One of the practical difficulties in operating bevel gears of the nature de- 
scribed is the difficulty of holding them so that they will be in proper contact; 
longitudinal motion in either shaft throws them out of pitch. The most 
serious problem, however, is in securing and maintaining shafts so that the 
extended axis lines of same pass through a common point. The effect of 
power transmission from pinion to gear is to put these axis lines out of posi- 
tion, moving them in opposite directions and resulting in end contact of teeth 
and concentrated load instead of evenly distributing the load along the whole 
length of tooth. In this particular the question of maintaining bevel gears 
is decidedly more of a problem than that of spur gears. In this latter case 
small end motions of carrying shafts produce no effect, while the wearing of 
bearings is only the shifting of pitch line, and, as it occurs slowly, it will, 
within reasonable limits, adjust itself. 

As a matter of further interest, I will mention that in this same room with 
these gears are three other sets of bevel gears having cut-steel pinions and 



SPEEDS AND POWERS 89 

mortise wheel with cast-iron rims. The diameters and ratios of these — 
speeds, mountings, and service — are practically the same as those described 
but the transmission of power is 1,100 instead of 1,300 horse-power. The 
pinions have 33 teeth, 53>^^-inch pitch, with 20-inch face, the teeth being 
planed down to 23^^-inch thickness on pitch-line. The wheels have 43 teeth. 
These gears have been in service some seven years. None of the pinions has 
ever given way; the wooden teeth in the wheels, however, last only from 
six weeks to two months, an extra rim being kept on hand for refilling and 
replacing. 

Mr. Lewis, in continuing the discussion, said that in regard to the pressures 
carried by gear teeth, Mr. Christie seems to lay down a rule making the 
product of speed and pressure constant. This would reduce the load in 
proportion to the speed, and it seems to me an open question whether that 
should be adhered to or not. I do not think it has been demonstrated how the 
pressure of the teeth should vary with the speed. Some experiments, I think, 
should be made which would indicate that more clearly than has heretofore 
been done. It is interesting to note his remarks regarding the influence of 
the hardness of the metal upon the pressures carried, and instead of reckoning 
the pressure by the inch as so much per inch of face, it seems to me the pitch 
should also be included, because the face of a gear tooth is very much like a 
roller, and the pressure carried by a roller varies with its diameter as well 
as with the face. Some authorities seem to think that it should vary with 
the square root of the diameter, others directly with the diameter, and I am 
inclined to the latter opinion. If gear teeth are proportioned for strength, 
they are also proportioned for wearing pressure and surface to carry the load. 

Mr. W. Trinks said: I wish to call attention to an article on high speed 
gearing in the November and December numbers of the "Zeitschrift des 
Vereins deutscher Ingenieure," 1899, by the chief engineer of the General 
Electric Company, at Berlin, Germany. The experiments show that there 
is no rule for the relation between pressure and speed, it depends upon accu- 
racy; the load on the teeth may be the higher the more accurately the gears 
are made. A remarkable method of manufacturing gears was the outcome 
of the experiment. The curves are laid out on paper three or four times the 
size of the real tooth, reduced to proper size by photography, transferred on 
sheet steel, and etched in. Thus the highest degree of accuracy is obtained. 

It was found that neither cycloidal nor involute curves gave the best 
results. Another curve was developed with a view to reducing the sliding 
motion between the teeth. The article contains very interesting diagrams 
on this point. By dividing the length of two working teeth into an equal 
number of parts, the amount of sliding action can be determined and the 
fact shown that it is reduced to a minimum by these methods. 

Another thing shown by the paper is never to place a flywheel close to a 
gear. If possible, have a good length of shaft between. Slight inaccuracies 
in the pitch of the wheel require acceleration or retardation of the mass. 



90 AMERICAN MACHINIST GEAR BOOK 

and in order to do this force is needed. This force causes a hammering on 
the teeth which may break them — in other words, plenty of elastic material 
should be between the inertia masses and the gears. I feel pretty sure that 
all engineers will be much interested in the article ; it is a valuable treatise on 
highspeed gearing. 

Mr. Christie added: The bevel gears described by Mr. Graves are very 
interesting and useful as a record. It is much more difficult to obtain satis- 
factory results with bevels than with plain spurs, as any deviation from cor- 
rect alignment is fatal to correct tooth action in the former. In this instance, 
while speed is very high, the pressure on the teeth is comparatively low — 
about 750 pounds mean pressure per inch of face. Thus the products of 
speed and pressure in relation to the pitch are considerably below the quantity 
assumed as a safe maximum. 

Regarding the quality of the material, the manganese is too high. While 
steel of this composition would be moderately hard and wear fairly well, it 
would be somewhat brittle. It is not surprising to learn that some teeth gave 
way by fracture. If the relative proportions of carbon and manganese in the 
steel were reversed, it would be a much better material for the purpose. 

A SPUR GEAR ANGLEMETER* 

In the design of spur gears it is desired to give to the teeth such a form that 
as the gears mesh with each other, the relative motion of the two will be the 
same as that of two cylinders whose diameters have the same ratio one to the 
other as have the diameters of the two pitch circles. By the aid of kinematics 
gear teeth can be so designed as to give exactly this relative motion between 
the gears. However, in the manufacture of gears, factors enter which make 
the form of the teeth of the gears as they come from the shop somewhat differ- 
ent from that developed by kinematics. This variation is quite marked in 
the case of rough-cast gears. Here several errors enter. Because of the 
difficulty and time required in developing a tooth outline according to kine- 
matics, arcs of circles which approximate the correct outline are used in laying 
out the gears in the drafting room. From these slightly inaccurate drawings 
the patternmaker makes the patterns, which are apt to vary slightly in form 
and the spacing of the teeth from the drawing furnished by the draftsman. 
These inaccurate patterns then go to the foundry, and from them the molds 
are made. In making the mold, in order to draw the pattern it is rapped 
loose, so that the mold is slightly larger than the pattern and still more 
inaccurate in outline. The casting is poured, and in cooling is warped out of 
shape because of the cooling stresses leaving a gear with a final error in the 
form of its teeth made up of several smaller errors, as just enumerated. 

* W. M. Wilson, American Machinist, April 13, 1905. 
Note. — The anglemeter curves for modern well cut generated gears show much smoother 
lines, and curves of gears rolled by molding generation — Anderson Process — approach 
closely the ideal of a straight line. 



SPEEDS AND POWERS 91 

A realization of the presence of these errors suggested that a knowledge of 
the final inaccuracy in the forms of the teeth of rough-cast gears would be of 
interest and perhaps not without value. With this idea in mind, an angle- 
meter for determining the variation in the angular velocity ratio of gears was 
designed and built. 

The anglemeter consists mainly of a responsive frame carrying a drum on 
which is wrapped a card. If these gears operate without vibration the 
pointer would draw a straight line around the card. Figs. 66 and 67 will be 
practically self-explanatory. The pointer was found to multiply the actual 
variations 9.4 times. 

TEST OF A PAIR OF CAST GEARS 

The instrument was connected to a pair of spur gears, No. 12020, each 
having a circular pitch of i}/^ inches and 20 teeth. Each gear was mounted 
on a 2 15^ 6 -inch shaft. As the two gears and the two shafts were the same 
size, no reducing motion was needed. The guides of the instrument were 
not long enough to allow a complete revolution of the gears, so the teeth were 
numbered from i to 20 and one card drawn for teeth i to 1 1 and another for 
teeth from 1 1 to i . The points on the curves corresponding to the time when 
these teeth came into contact were marked and then the cards were cut to 
these marks and pasted end to end as shown in Fig. 66. 

The details of the design of this pair of gears were not known, but judging 
from the shape of the teeth they were modifications of involutes and the out- 
lines of the teeth were probably laid out according to some empirical method 
in general use in the manufacture of rough-cast gears. Before the test the 
gears had been run long enough for the bearing surface to become smooth. 

Cards were taken when the gears were adjusted at different distances 
apart. If the gears had been true involutes the velocity ratio should be 
constant and the same for the different distances between the centers. Whe- 
ther the velocity ratio were constant or not is readily seen from the curves in 
Fig. 66. Due to the uncertainty of the shrinkage of castings, the diameter 
of the gears was not exactly equal to the computed diameter, but a trifle 
greater. Judging from the clearance at the ends of the teeth the proper 
distance between the centers of the gears was 9^3 inches. Card A, Fig. 66, 
gives a curve corresponding to this distance between centers. Two curves 
were drawn before the card was removed and the ability of the instrument to 
duplicate a curve is considered as evidence of its accuracy. It is seen from 
the curve that the velocity ratio instead of being constant is quite erratic in 
its variations. While in general the variations in the curve are not related 
to each other in any way, yet for a portion of the card at least the waves in the 
curve correspond roughly with the teeth on the gears. This is especially 
true of the portion of the curve corresponding to the teeth from 1 1 to 20. 
The curves on Card B were drawn when the distance between centers was 
9^/'iQ inches. In this case there is a more marked relation between the waves 



92 



AMERICAN MACHINIST GEAR BOOK 



in the curves and the teeth on the gear than in the former case. For Card C 
the distance between centers was io}y{Q inches, and the regularity of the waves 
is still more marked. In this case the card was taken for half a revolution only. 

For all the cards the arrows above the different curves indicate the 
general direction of the tangent to the curves which give the angular accelera- 
tion of the driven as compared with the driver. 

The rough jagged nature of the curves portrays in a vivid manner the 
wide variance from that uniform positive motion of pitch surfaces necessary 
for securing high gear efficiency which exists in cast gearing. It also indicates 
that although the relatively rough profile surfaces of cast teeth are contribu- 
tory causes of inefficiency they are nowhere as serious as malformation and 
variations in gear tooth proportions. 




2Q>^ 



.3" ^ 

9 "/ifi between Centers 




10 /lo between Centers 
FIG. 66. CURVES FROM CAST GEARS. 

From the curves shown the following conclusions have been drawn relative 
to the forms of the teeth: 

(i) The fact that the relation between waves in the curves and the teeth 
of the gears is increased indicates that all the teeth are subject to a common 
error due to the empirical method in which the outlines of the teeth were 
developed, also that this error causes a greater variation in the velocity ratio 
when the distance between the centers of the gears is increased. (This is in 
accordance with the empirical method presented in Kent's hand book and 
attributed to Molsworth. By using the method referred to, the resulting 
tooth outline would be wider below and narrower above the pitch circle than 
true involute teeth. When the distance between centers is normal these 
errors annul each other almost completely, but as the distance is increased 
the result of the errors is more apparent.) 



SPEEDS AND POWERS 



93 



(2) Irregularities in the curves indicate errors peculiar to the individual 
teeth, which evidently are involved in the progress of patternmaking and 
molding. 

(3) A shifting of the general vertical position of the curves on the cards, as 
shown at teeth 4 and 14, indicates errors in the spacing of the teeth. 

An effort has been made to analyze the curve in Card A . The portion of 
this curve ABC has been chosen as being a wave whose relation to an indi- 
vidual tooth is the most evident. The abscissa AC represents the angular 
space occupied by one tooth on the gear, and AB and BC each represent one- 
half of that angular space. The ordinate between A and B measures 0.18 
inch, and that between B and C measures o.ii inch. Dividing by 9.4, the 
constant for the instrument, it is found that the driven gear gains on the 
driver by an angle whose arc measured on the circumference of the shaft is 
0.18 divided by 9.4 equals 0.019 inch, while the gear turns through an angle 

- 360 , . - 

equal to X >^ = 9 degrees. 



An angle whose tangent is 

0.019 



is 45 minutes. 



1.46885 (radius of shaft) 

Computing in the same manner the angle corresponding to the ordinate 
BC is found to be 30 minutes. That is, while the tooth No. 16 is in action 
the driven gains relative to the driver by an angle of 45 minutes while the 
latter is turning through an angle of 9 degrees, and then loses by an amount of 
30 minutes in the same space. An error of 0.019 inch measured on the cir- 
cumference of the shaft corresponds to 0.06 inch measured on the pitch circle. 

TEST OF SPECIAL GEARS 

The second pair of gears to which the instrument was attached consisted 
of a No. 1 2016 pinion having i3^^-inches circular pitch and 16 teeth and a 
No. 12050 gear having 50 teeth. The gears were designed to mesh properly 
when the distance between centers varies by an amount equal to 3^^ inch. 
The shrinkage of the casting was not as much as was expected, so that the 
normal distance between centers is 0.20 inch more than the sum of the com- 
puted radii. If the gears were true involutes the velocity ratio should be 
constant when the distance between centers of the gears varies from 15.7 to 
16.2 inches. 

The outlines of the teeth of the gears are involutes having an angle of 
obliquity of 233^ degrees when the distance between centers of the gears is 
normal. The mold for No. 12016 was made from a pattern, while for No. 
12050 it was made from a tooth-block as used in a Walker gear-molding ma- 
chine. The pattern and tooth-block were both made from drawings laid out 
in the drafting room. The tooth outlines for these drawings were developed 
according to kinematics, so that for the exception of any small error which 



94 



AMERICAN MACHINIST GEAR BOOK 



the patternmaker might make, the teeth on the pattern and tooth-block are 
correct involutes. 

How near the castings approached to involute gears is shown by the curves 
in Fig. 67. As the two gears were not of the same size, it was necessary in 




15.7 " between Centers 




16.2 between Centers 




16.45 between Centers 




16.82" between Centers 
FIG. 67. CURVES FROM SPECIAL GEARS. 

taking the cards to use some means of making the average velocity of the two 
carriages of the instrument the same. To do this the hub of the gear was 
turned to a smooth surface and the end of the shaft of the pinion was turned 
down until its diameter bore the same ratio to the diameter of the hub of the 
gear that the diameter of the pitch circle of the pinion bore to the diameter 
of the pitch circle of the gear. This gave a rigid and accurate reducing motion . 
The method of taking the card^ was the same as for the other pair of gears 



SPEEDS AND POWERS 



95 



except that a curve for a complete revolution of the pinion was obtained upon 
a single card. The distance between centers of the gears for the different 



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cards is as indicated in Fig. 67. Cards D, E, and F cover the range of varia- 
tion in the distance between centers for which the gears were designed. From 



96 AMERICAN MACHINIST GEAR BOOK 

these curves it is seen that there are no waves on the curves corresponding 
to the individual teeth, the curves for the most part are smooth, and are 
practically the same for the different distances between centers of the gears. 
For card G the distance between centers was made 16.45 inches, or 0.75 inch 
more than the least distance between centers at which the gears were supposed 
to run. The remarks made in regard to curves D, E, and F apply equally 
well to this curve also. For cards H and / the distance between centers was 
so great that the point of contact of the teeth did not follow the line of obli- 
quity throughout the angle of contact but lay outside of this line for the latter 
portion of the period of action of the teeth. 

This is clearly brought out in the forms of the curves in which waves cor- 
responding to individual teeth are quite evident. 

It will be noticed that for all of the curves for the second pair of gears there 
is an extended wave near the middle of the card and also near each end, so 
that if several cards were placed end to end so as to form a continuous curve 
there would be two complete waves on this curve per revolution of the pinion. 
This indicates that the pinion instead of being exactly round is oval in form. 
This deformation can be traced to the molder in the foundry who, in order to 
draw the pattern, raps it loose from the sand and unless great care is taken 
increases the diameter more in one direction than in the other. The wave in 
the curve is traced to the pinion instead of to the gear because the spacing of 
the teeth for the gear was done by means of a molding machine and any errors 
which might occur would be peculiar to the individual teeth instead of being 
common to a group of teeth and gradually increasing and decreasing from 
zero up to a maximum and then back again to zero, as indicated on the cards. 

From the curves in Fig. 67 the following conclusions are drawn relative 
to this pair of gears : 

1. The outlines of the individual teeth are true involutes. 

2. The spacing of the individual teeth on both the gear and the pinion is 
accurate. 

3. The pinion is slightly oval in form. 

4. For the gears used the angular velocity ratio is constant through a 
range in the variations in the distance between centers equal to 0.75 inch. 

GENERAL CONCLUSIONS 

If one would be allowed to draw conclusions from the tests of two pairs 
of gears only, the following statements might be made in regard to the errors 
in the teeth of rough-cast gears: 

1. Of all the errors the greatest is due to laying out the teeth in the draft- 
ing room. 

2. Given an accurate drawing the patternmaker can make a pattern very 
accurate in form. 

3. In using the patterns the molder is apt to make the mold slightly oval 



SPEEDS AND POWERS 97 

in form, but in using a molding machine the error induced in the foundry is 
very small. 

4. The surfaces of the teeth of cast gears are so smooth as not to affect the 
angular velocity ratio. 

The difference in the appearance of the curves for the two pairs of gears 
is very much in accordance with the manner in which they meshed, the second 
pair running with very much less noise than the first pair. At a speed of 250 
revolutions per minute the first pair almost threatened to shake the testing 
machine to pieces, while at the same speed the second pair ran with but little 
vibration. 

It might be said that the patterns for the second pair of gears were com- 
pleted before there had been any thought of subjecting the gears to a test, 
which fact eliminates the possibility of unusual care having been taken in 
making the patterns. 

COMMENTS ON GENERAL CONCLUSIONS 

Although the foregoing conclusions are based upon the behavior of cast 
gears made without the exercise of any exceptional care in their founding, 
an analysis of these conclusions will throw considerable light on the behavior 
of cut and generated gearing, indicating as conclusively as the most exacting 
mechanical tests the conditions in gear fabrication which should be given 
most attention. In the first place, it is quite evident that the uniform posi- 
tive motion of pitch surfaces essential for a constant angular velocity ratio is 
primarily a matter of uniformity in general tooth section and evenness in 
tooth spacing. In the second place, it is evident that if a constant angular 
velocity ratio of gears could be attained, the exact form of tooth would be 
of quite secondary importance. 

The modern process of gear cutting, generation, takes care of the uni- 
formity in general tooth section, as each tooth cut on a generated gear is, so 
far as the mechanical precision of the generating machine permits, a replica 
of every other tooth and its form is a conjugate of the generating cutter tooth. 
The efficiency of generated cut gears is, consequently, dependent upon the 
constancy of the angular velocity ratio between the generation cutter tool 
and that of the work arbor carrying the gear blank to be generated, and the 
rigidity with which the gear blank is held on the work arbor. The vital 
difference between machining generation and molding generation is a 
question of rigidity of gear blank. Machining generation being of necessity 
an intermittent process, in order to permit the advance of the gear blank, the 
gear blank is during generation under a series of intermittent stresses which 
have a tendency to cause an angular vibration in the gear blank. In mold- 
ing generation, on the other hand, the stress on the gear blank is a more uni- 
form one and the tendency to cause angular vibration in the gear blank is 

absent, so the constancy of the angular velocity ratio of hot rolled gears is 
7 



98 



AMERICAN MACHINIST GEAR BOOK 



dependent almost entirely upon the constancy of the angular velocity ratio 
between the heavy timing gears which actuate the machine. 

GEAR EFFICIENCY 

No thorough investigation has yet been made of the efficiency of gears, 
and very few data of any sort has been published, therefore Uttle can be said. 
One thing is tolerably well known : The efficiency of a gear drive varies with 
its ratio — that is, a reduction of 2 to i will be more efficient than one of 10 
to I , other features governing the efficiency being the accuracy with which the 
teeth are formed, and spaced, the arc and obliquity of action, and the condi- 
tion of the engaging surfaces. It is generally conceded that the length of 
face does not affect the efficiency. 



2 2>2 degree 


involutes. 


24^4 degree involutes. 


19K degre 


e involutes. 


16" between centers. 


16.2" between centers. 


15.7" between centers. 


Tangential 


force at pitch 


Tangential 


force at pitch 


Tangential force at pitch 


line, 36 


I lbs. 


line, 


355 lbs. 


line, 


353 lbs. 


R. P. M. 


Efficiency, % 


R. P. M. 


Efficiency, % 


R. P. M. 


Efficiency, % 


88 


92.40 


116 


91.00 


no 


90.00 


123 


90.00 


68 


96.90 


69 


94.40 


140 


89.90 


90 


91.50 


127 


86.80 


156 


91.30 


140 


89.90 


147 


88.90 


172 


91.30 


163 


90.10 


162 


88.80 


188 


92.30 


177 


90.00 


196 


88.90 


216 


92.30 


197 


99.90 


218 


89.20 


229 


92.40 


208 


90.00 




Av. 89.60 




Av. 91.50 


214 


90.30 
Av. 91.20 






Efficiency of Gears 


Efficiency of Gears 


Efficiency of Gears 


alone, 


92.30% 


alone, 


92.10% 


alone 


90-50% 



Table 16 — Gear No. 120 16 Driving 12050 — Lubricated 

This loss comes mainly through the sliding action of the teeth when 
entering and leaving contact; it would seem, therefore, that the arc of action 
should be as short as possible, carrying the load of that portion of the tooth 
where it can be best borne, also, as there is less friction in the arc of recess 
than in the arc of approach, it will increase the efficiency to lengthen the 
addendum of the driving gear. The contact, however, should be for an 
equal distance above and below the pitch line to avoid extreme sliding 
friction. If the entire action was rolling contact, there w^ould be Httle loss. 

Tables 15 and 16 were originally published in American Machinist, by 
W. M. Wilson. They give the result of experiments with the same case 
tooth spur gears mentioned on pages 98 and 100. The gears in Table 16 
were made with extra long teeth, the various angles of obliquity as given were 
obtained by adjusting the gear centers. 

The result of these tests is summed up as follows: 

I. The efficiency of rough gray iron spur gears is independent of the speed 
of the gears within the range covered by this report. 



SPEEDS AND POWERS 99 

2. There is no indication from the Umited number of tests that the 
amount of power transmitted affects the efficiency to any great extent. 

3. The use of a heavy grease on the teeth increases the efficiency sUghtly 
(average from tests 1.7 per cent.). 

EFFICIENCY OF LARGE GEARS 

Many of us know things that are not so, and with some of us a part of this 
useless knowledge may be in regard to the efficiency of large gears. 

We believe that we are well within the bounds of truth in saying that the 
majority believe that large gears are very inefficient. 

This misinformation or lack of information is easily explained. The 
data that are available in regard to the efficiency of gears of any size are few 
at best and apply to pinions and small gears. Large gears have not been 
extensively tested, for there are but few technical schools and factories 
equipped with appliances for gear testing and capable of absorbing several 
thousand or even several hundred horse-power. Again we do not always 
distinguish between a large gear with cast teeth and a similar gear with cut 
teeth. 

In the early days of factory engineering, gear drives were common for 
transmitting power from shaft to shaft throughout the plant. As time went 
on these drives were replaced — in some cases by belting, in other cases by a 
change from mechanical transmission to electrical distribution of power. It 
has been very easy to assume that the reason for discarding the gears was 
because of their inefficiency. Enthusiastic advocates of electric drives have 
time and again referred to the substitution of motors for mill gearing with 
elation and have either stated or implied that the change brought about a 
great saving of power. 

In further support of the common belief that gearing in large sizes is 
inefficient we quote the following from the presidential address of Mr. Denny 
to the Institution of Marine Engineers, in England, on October 5, 1908: 'Tt 
has frequently been suggested that if some inspired engineer could evolve a 
system of gearing that would be lasting and reliable, not too noisy, and would 
not absorb in friction more than, say, 10 per cent, of the power, turbine engines 
would be capable of application to any speed of vessel and to any size of 
propeller." Here Mr. Denny gives expression to an oft-repeated suggestion, 
that if large gearing could be made having an efficiency of 90 per cent., a big 
step forward would be made. 

With direct bearing on this belief we have the record of performance 
of the largest gears in point of horse-power ever made and tested — the 
Melville- Macalpine reduction gear, through which 6,000 horse-power has 
been transmitted. 

To show the measure of the performance we quote a paragraph from an 
article by George Westinghouse in the January number of The Electric 
Journal: ''Considering the important bearing of the question of efficiency on 



100 AMERICAN MACHINIST GEAR BOOK 

the ultimate success or failure of the gear, it is peculiarly gratifying to have 
found by repeated trial and careful measurement that the transmission loss 
hoped for by Mr. Denny has been divided by seven. To be exact, the 
efficiency surpasses the more than satisfactory figure of 98.5 per cent." 

The hope was for an efficiency of 90 per cent., the fulfilment was an 
efficiency of 98.5 per cent. 

Large bevel-gear drives have perhaps been especially condemned; it is, 
therefore, of interest to read the following from a paper presented by Prof. 
C. M. Allen to the American Society of Mechanical Engineers on the testing 
of water wheels: ''The total horse-power delivered to the generator was 
approximately 700. The driving gear was of the ordinary wood-mortise 
type, outside diameter 6 feet 5 inches approximately, with 68 teeth 14 inches 
wide, meshing with a cast-iron pinion which had 48 teeth with planed-tooth 
outline. At full load the loss of the gear was 3.5 per cent, and 3.4 per cent, for 
two separate units, or the efficiency of the horizontal-shaft vertical-wheel 
gear drive was about 96.5 per cent. The gears were well lubricated with a 
thick grease. 

"About nine months later it was necessary to test one of these same units 
in exactly the same manner. The loss in gears this time was a trifle less, the 
test giving 3.1 per cent. 

"All of the information obtained concerning the loss due to bevel-gear 
drives leads the writer to conclude that if gears are properly designed, set up, 
and operated, and are not overloaded intermittently or continuously or left 
to care for themselves, they should show an efficiency of from 95 to 97 per 
cent." 

Are not these bevel-gear drives efficient compared with the majority of 
mechanical devices? 

As a matter of fact, do not the above figures agree with common sense? 
The quantity of heat generated by the absorption of one horse-power for an 
hour is 2,545 British thermal units. If we try to give an expression to the 
statement that large gears are inefficient by assuming a loss of 10 per cent., 
in the case cited by Professor Allen, 70 horse-power would be dissipated in 
heat. A multiplication will show what this means in British thermal units 
per hour, and the presumption is strong that the gears would heat, cut, and 
wear under such a condition. 

The same reasoning will probably apply to many large-gear drives con- 
cerning which there has been speculation in regard to efficiency. Had they 
been extremely inefficient, they would not have operated satisfactorily for a 
long period of time. 

These figures show that large-gear drives are not necessarily ineflScient, 
but, on the contrary, may be decidedly the reverse. 

COMMENTS 

As the efficiency of gearing depends upon the uniformity of the angular 
velocity of the pitch surfaces, it follows that the finer the pitch of the gears 



SPEEDS AND POWERS lOl 

and the more numerous the teeth, the more uniform will be the angular 
velocity of the pitch surfaces and the better the operating efficiency of the 
gears. Consequently, for any given pitch the efficiency will increase with 
the number of teeth. This fact explodes the popular belief that large gears 
are inefficient. They are, on the contrary, apt to be relatively more efficient 
than smaller gears, providing they are accurately mounted and well aligned, 
as customarily their pitch is proportionally finer in comparison with their 
diameters than in the case of small and relatively coarser gears. The inac- 
curacies which cause variations in angular velocity of pitch surfaces become 
less potent as the number of teeth increase. 



SECTION IV 
Gear Proportions and Details of Design 

These formulas should not be used indiscriminately, as one design will not 
meet all conditions. No attempt has been made to proportion arms or rim 
to the power to be transmitted, as all proportions are derived from the pitch 
and face of the gear, with the double object of obtaining equal strength and 
sound castings. 

As the smallest section of a casting is, per square inch, its strongest part, 
this fact should enter more into the question of proportion than has been the 
custom in the matter of gears. To illustrate the value of this, a test piece cut 
from the point of a cast gear tooth will often be as much as 30 per cent, 
stronger than a similar piece cut from its root. Hence the rim of a gear is 
made thinner than has been the practice, and the central rib deeper, to secure 

Taper V2 Inch 

per Foot, -r- 




FIG. 68. PLAN AND SECTION OF A SPUR GEAR, SHOWING NOTATION. 



the necessary section and thus obtain a stronger casting with the same weight 
of material. This same rule applies to the hub, the outside diameter being 
reduced and a deeper rim added, and reinforcements placed over keyways, or, 
better still, two reinforcements directly opposite, especially if the gear is to 
be balanced; in fact, this is imperative even at relatively low velocities. For 
the same reason the teeth should be cored whenever it is possible to do so, 
as the rim of a casting for a cut gear has always the heaviest section, and, 
therefore, most subject to blow holes, especially at the junction of the arms 



102 



GEAR PROPORTIONS 103 

and rim. It follows, then, if this be true, that the more uniform the section 
throughout, the sounder and stronger the casting. 

Cored teeth, however, should be machine spaced, as uneven spacing will 
render it difficult, if not impossible, to cut the teeth in the usual manner. To 
secure accurate spacing when cutting, it is absolutely necessary that the 
cutter have an equal amount of stock to remove from each side of the tooth 
space. Also exposure to hard scale and core sand quickly destroy the cutter. 
When the teeth are to be planed this point is not of so much importance, as a 
little unevenness of stock can be readily taken care of, but the cut should be 
always under the scale if time is of any importance. 

FORMULAS 

These formulas are based on Brown & Sharpe standard 143^ -degree 
involute tooth. 

Thickness of Rim, M = ^^^^, or 1.2 s p' 
' p -^ ^ 

Mean Thickness of Rim, M' = — — , or 1.60 p' 

P 

Mean Thickness of Rim under Tooth, R' = ~ , or 0.913 p' 

Whole Depth of Tooth, W = ^^^, or 0.6866 p' 

P 

Minimum Thickness under Tooth, R' = , or 0.563 p^ 

Area of Rim Total, MF' = M'F 

Area of Rim under Tooth = R'F 

Average Area of Arm, ^ = /F 1.3, or MF 0.52 

\M I 27 
Average Thickness of Arm, A = a / '—^ 

Average Width of Arm, E = t^A 

Outside Diameter of Hub = Bore + %\^NF 

Number of Arms = 4, 6, 8, 10, etc., according to design and diameter of 
gear. 

p = Diametral pitch. 

p' = Circular pitch. 

t — Thickness of tooth at pitch line. 

DISCUSSION OF FORMULAS 

Thickness of Rim, M — The thickness of rim should be equal to 3.927 
divided by the diametral pitch, or 1.25 multiplied by the circular pitch. 
When the gear is small and accurately made it is often good practice to make 



I04 AMERICAN MACHINIST GEAR BOOK 

the dimension 1.12 of the circular pitch, and so secure the same section and 
additional strength by adding 50 per cent, to the depth of the central rib. 

Mean Thickness of Rim, M' — By mean thickness is meant the thickness 
of one side of a parallelogram necessary to contain the actual area of the rim. 
This will take care of the central rib and fillets, so that by multiplying the 
width of the face of the gear by dimension M' the entire area of rim may be 
obtained. This will be found necessary also for estimating the weight. 

Mean Thickness of Rim under Tooth, R' — This dimension ( ' I multi- 
plied by the width of the face will give the area of the entire section of the rim 
and rib under the teeth. 

Minimum Thickness of Rim under Teeth, R — This determines the thick- 
ness of the rim under the tooth measured at the edge of the rim. 

Whole Depth of Tooth, W — This gives the whole depth of the tooth as 

2. 1 o 
per Brown & Sharpe standard = 0.6866 p' , or — —' 

Total Area of Rim — The total area of the rim is found by multiplying the 
mean thickness M' by face of gear F. 

Area of Rim under Tooth — The area of the rim under the tooth is deter- 
mined by multiplying the mean thickness R' by the face F. 

Average Area of Arm, M — The average area of the arm is that area mid- 
way between the inside of the rim and the outside of the hub, and is found by 
adding 30 per cent, to the area of the tooth at the pitch line, or the thickness 
of tooth at pitch line / X -F X 1.3. The same result may be reached by tak- 
ing 0.52 of M' X F, although the foregoing is simpler. Taper of arm to be 3^^ 
inch per foot above and below this point. 

Average Thickness of Arm — The average thickness of arm {A ) may be de- 
termined by dividing the quotient of the area of arm jE X 1.27 by 3, and 
extracting the square root. If the arm was made in the form of a parallelo- 
gram it would not be necessary to multiply the area by 1.27, but as it is to 
be elliptical, this is essential to insure sufficient section, as 27 per cent, of the 
area of the parallelogram is lost v/hen inscribing an ellipse there in. 

Note. — If width of arm is desired, 2 or 2^ times the thickness instead of 3 times, as 
given, 2 or 23^ is to be substituted. 

Average Width of Arm, E — To determine the width of an arm multiply 
its thickness by 3. 

Outside Diameter of Hub — As a rule the outside diameter of a gear hub 
is made double that of its bore, but when keyways are reinforced, or the gear 
is to carry a load less than proportional to the diameter of its shaft, the hub 
diameter may be less than this rule prescribes. Thus, if a gear of 30-inch 
pitch diameter was mounted on a shaft 15 inches in diameter, the hub diam- 
eter should only be increased sufficiently to maintain its section and strength 
proportional to the gear, not to the shaft. The formula given will propor- 
tion the hub to easily carry the entire load applied to the gear but should be 



GEAR PROPORTIONS 



105 



used with discretion. Generally, however, when the bore is small or propor- 
tioned to the diameter of the gear, the outside diameter of the hub may be 
taken as 1.75 times the diameter of the bore with reinforcement for key ways. 



1.45 



1.40 



1.35 
I 1.30 

H 

«H 

<=> 1.25 

£ 1.20 



u 

.^ 1.10 

0. 

"3 1.05 
1.00 
0.95 
0.90 



IT L 

J ±^ 

IT I JL ^ 

____--_-_______ ________________________--_______-_______-.______^ J _j_ 



30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 100 200 210 

Number of Teeth 
CHART 5. MULTIPLIER FOR INCREASED NUMBER OF TEETH OF SPUR GEARS. 

From the foregoing it is obviously almost as important that the hub 
should not be disproportionately heavy as it should be heavy enough. But 
if for any reason a materially heavier hub is required, it should be split, by 




160 Teeth. 
3 Inch Pitch 
20 Inch Face 

FIG. 69. PLAN AND SECTION OF SPUR GEAR WITH SPLIT HUB. 

means of thin cores, into as many equal, radial sections as there are arms in 
the gear, thus obviating blow-holes, and strains caused by shrinking. Fill 



io6 



AMERICAN MACHINIST GEAR BOOK 



the cored spaces with babbitt or lead before machining, and shrink steel 
bands on the hub for a grip on the shaft. See Fig. 69. 

Number of Arms — There is no definite rule for this, as this point depends 
almost entirely upon the judgment of the designer. In general, however, 
gears up to 60 inches are either webbed or with four or six arms to suit condi- 
tions; over 60 inches eight arms are generally used and over 80 inches in 
diameter 10 arms. In no case should the greatest distance between the 
arms exceed the length of the arm measured from the center of the gear to its 
intersection with the rim. 

Width of the Face — The face of spur gear is generally estimated at two or 
three times its circular pitch, as follows: 



1 diametra 
1 3^^ diametra 
1 3^-2 diametra 
1% diametra 

2 diametra 
2\'2 diametra 

3 diametra 
diametra 
diametra 
diametra 
diametra 
diametra 
diametra 
diametra 
diametra 

18 diametra 
20 diametra 



4 

5 

6 

8 

10 

12 

14 
16 



pitch . 
pitch . 
pitch, 
pitch . 
pitch, 
pitch, 
pitch . 
pitch . 
pitch . 
pitch . 
pitch . 
pitch, 
pitch . 
pitch, 
pitch . 
pitch . 
pitch . 



7K 



5 

4 

3 
2 



y-i 



inches face 
inches face 
inches face 
inches face 
inches face 
inches face 
inches face 
inches face 
inches face 
inches face 
inches face 
inch face 
inch face 
inch face 
inch face 
inch face 
inch face 



It is becoming better understood, however, that a wider face is more effi- 
cient ("increasing the face does not increase the friction of the teeth in pro- 
portion"),* and as the wear of the teeth is governed by the diameter of the 
gear, or rather by the combination of diameters and the width of the face, 
the face, therefore, should be amply wide, and the pitch just sufficient to resist 
fracture. 

Street-railway gears are made 3-pitch, 5-inch face, with good results. A 
gear face of five times its circular pitch is now generally considered to be good 
proportion. 

WEBBED SPUR GEARS 

No definite rule can be laid down for the design of webbed gears. See Fig. 
70. Generally the thickness of the web is made equal to E!' , which is the 
thickness of the rim at its thickest part. 

* George B. Grant. 



GEAR PROPORTIONS 



107 



Core holes U tend to make a sounder casting, and furnish means to secure 
the gear while machining. When these holes are made large, or are shaped to 
follow outlines of the arms and rim, ribs are added on each side. Care 




FIG. 70. PLAN AND SECTION OF WEBBED SPUR GEAR. 

should be exercised, however, not to make the arm too light, for when the hub 
is heavy the light section connecting the rim and hub will cool too rapidly, 
setting up serious shrinkage strains, and causing flaws in the casting that 
cannot be remedied by annealing. Sharp corners, small fillets, and narrow 
ribs should be avoided for the same reason. 

SPLIT SPUR GEARS 

It is not good practice to split a gear between the arms, but when this is 
necessary, the following points should be kept in mind (see Fig. 71): 

Bolts should be placed as close to the 
rim as possible. 

The dimension h must in no case be 
less than the dimension of a; otherwise the 
bolts will be subject to other than the 
direct tensile stress tending to spread the 
gear. 

Section C-C should be stiff enough to 
resist any strain tending to bend the 
lugs. By placing the bolt close in the 
corner of the rim and the lug, the length 
of the lug may be reduced, as this lug need 
only be long enough to counteract leverage 
on bolt. 

The bolts should be sufficiently heavy to carry the load applied at the 
pitch line of gear tooth, not neglecting the initial stress set up by the tighten- 
ing of the nut, which is generally neglected, sometimes disastrously. When a 



A-' 


y 


n 


t 


/' 


^ Uy 


1* 


— -- 




31 


1\ 



Fig. 71 




DIAGRAM ILLUSTRATING SPLITTING OF 
GEARS AT THE RIM. 



io8 



AMERICAN MACHINIST GEAR BOOK 



bolt is placed close in the corner it is necessary to use a stud bolt, drawing up 
the nuts as the gear halves are brought together. 

When the dimension h is shorter than a, Fig. 71, as is generally the case, 




w, 



wmwm, 



m 



ko 



69 Teeth. 
3 Inch Pitch. 
5 Inch Face. 



o 



o 



O/T* 



FIG. 73. PLAN AND SECTION OF AVERAGE DESIGN OF SPLIT RAILWAY GEAR. 

the load on teeth of gear will cause a fracture of the rim, as illustrated in 
Fig. 72. 

The best method of splitting gears is through the arm, as illustrated in 







Iq![o 



I 



^T^MM 



tm 



FIG. 74. PLAN AND SECTION OF AVERAGE DESIGN OF LARGE SPLIT GEAR. 

Fig. 73, which is a cut of the type used on street railways. When the gear is 
large, Fig. 74 illustrates a good average design. 

When splitting a gear of an odd number of teeth, the split should be made 
y^ of the circular pitch off the center line. This will bring the split through 
center of two tooth spaces. It is good practice to spline the adjoining sur- 



GEAR PROPORTIONS 



109 



faces, as illustrated in Fig. 73, instead of using fitted bolts or depending on 
dowel pins. A spline % inch wide and }i inch high will answer for any but 
the largest gears. 

One point that must be considered in designing split gears for high speed is 
the fact that weight at any point or part of the rim, and not integral to the 
rim proper (as lugs for bolting, see Fig. 71), locates the bending moment, and 
if safe speed is exceeded to the moment of fracture, it will occur at or near 
such weight. 

This fact is pointed and explained by Charles H. Benjamin, in an article 
in the American Machinist of December 26, 1901, entitled "The Bursting 
of Small Cast-iron Flywheels." 



I-SHAPED ARMS 

For gears such as are shown in Fig. 69, the proportion of rim, arms, and 
hub may be determined according to formulas given above, the area of arms, 
of course, being contained in an I instead of an elliptical section. For gears 
of this size, however, there is more variation in 
the design of the gear, and the values given 
cannot be followed so closely. 

The I-section arm is much more desirable 
in heavy gears because it distributes the metal 
contained in a section of the arm over a greater 
surface of the rim at their intersection than 
does the elliptical arm, and, therefore, lessens 
liability of blow holes at this point. Aside 
from this, it gives better support to rim in 
case the face is wide, and makes a stronger 
section than the elliptical arm of the same weight. 

For the above reasons I am disposed to advocate a connecting-rod section, 
such as is illustrated by Fig. 75, for smaller gears. 




FIG. 75. SUGGESTED CONNECT- 
ING-ROD SECTION. 



CONNECTING-ROD-ARM SPUR GEAR 

This design could be applied to gears of all sizes: As the face increased, 
the section could be changed — as per dotted Hues. This, however, would 
allow a central rib instead of two ribs on each side of the rim as when the ^ 
is turned the other way (see Fig. 69), but would allow the use of bolts instead of 
links when a spHt gear had no hub projections. This design is illustrated by 
Fig. 76. 

For gears of an extremely wide face it will be found that the formula for 
arm will give a section that cannot be contained in the space between hub and 
rim. This practically means making a web gear. However, when the face 
is wide it would seem better to use a light web, say, according to dimension 



no 



AMERICAN MACHINIST GEAR BOOK 



on Fig. 70 and extending ribs toward the side, making a cross-sectioned arm, 
or using the section shown by Fig. 69. This section is generally the most 
desirable, but this depends upon conditions. 

The formulas given here will form a basis for the design of worm and bevel 
gears, although I beheve that the + or cross-shaped section is superior to the 
oval arm for worm gears on account of the side strain encountered. 




I o 



80 Teeth, 6 Inch Pitch, 14 Inch Face, 15^ 
Inch Bore, Cast Steel 27,000 Pounds, Rough 
Weight 23,000 Pounds with Cored Teeth. 

FIG. 76. PLAN AND SECTION OF LARGE SPUR GEAR WITH CONNECTING-ROD ARMS. 



FOR CALCULATING WEIGHT 

The accompanying Chart 6 gives a rapid, approximate method of 
calculating the weight of a cast-iron spur gear blank for a cut gear, designed 
according to the above formulas. 

This table was derived from a formula by Reuleaux, which gives the 



N 





2 


■4 


6 


8 


20 


5-04 


5.60 


6.18 


6.77 


, 7-38 


30 


7-99 


8.61 


9.24 


9.89 


10.52 


40 


11.09 


11.90 


12.59 


13-30 


14.02 


50 


14.74 


15.48 


16.23 


17.00 


17-77 


60 


18.55 


19-35 


20.15 


20.97 


21.80 


70 


22.65 


23-50 


24.36 


25.24 


26.12 


80 


27.02 


27-93 


28.85 


29.79 


30-73 


90 


31.69 


32.66 


33-63 


34.62 


35-63 


100 


36-63 


37-67 


38.70 


39-75 


40.81 


120 


47.40 


48-54 


49.69 


. 50.85 


52-03 


140 


59-30 


60.56 


61.82 


63.10 


64.27 


160 


72.35 


73-73 


75-10 


76-39 


77-90 


180 


86.54 


88.03 


89-52 


91.02 


92.54 


200 


101.88 


103.48 


104.98 


106.70 


108.34 


320 


118.36 


120.08 


122.15 


123-52 


125.27 



Example: A gear 50 teeth 2 inches pitch, 4 inches face, we have: bc^ 14-74 
14.74 = 235.84, say, 236 pounds. 

Weight of Cast-iron Gearing. Reuleaux. 



4X 22X 



GEAR PROPORTIONS 



III 



weight of a cast-tooth gear from the combined product of a constant for the 
number of teeth, face and square of the circular pitch, as follows: 

"The approximate weight of gear wheels, W, may be obtained from the 

following: 

"W = 0.0357 bc^- (6.25A' + o.o4xV-), where & = face, c = circular pitch, 
N = number of teeth, and W = weight of gear. 

''The following table will facilitate the application of the formula; it gives 

the value ^-^ for the number of teeth which may be given, and the weight may 

DC" 

be readily found by multiplying the value in the table by bc^:'' 



4' ~ """ 1 ■"" £' Z"' ~ 




<^ ^^ 


'' -«'' ^ 


Q3/. ^' -^K'^ 


av4 ^^p- 


^ ^•^'' 


' ^-^ 


•' ^- 


^1/ e2^^ 


3J-2 ^5?' 


T ^ 


^^ 




S14 -^ 




/* 


V' 


2'" 


0" -. \4' 


3 y/' 


^ 


7t ± 




93/. Jw 


^ //^ 


/ 


7/^ 


/^ 


9i'^ y 


<:i 2 >> 


m -.^" 


S - X ^ -^ 


^ ., y<^ i^ 


"ol/ v^ 




^ >« 


£ f 


X t 


^9" 1^ 


a,^ r 


Ph ^ 


>^ ^ 


^ „ 6 


3l^ A 






/ /-<lil.JJ?T-ii' 




lU -* 


^ ^'^ p'2-158 K 




t 




■\\/. '~ Jl Weight of Spur Gear Blank (in Pounds ) — 






jt , , !• i /-I i T -m 1 


,r ihe above applies to Cast Iron iilanks. 






7r 


- X ^^ 


fj 


?i ^* 


/4 -* 




T 


11 w 


V7 IS -(- 


I«^ 


^l 


it 


» It 


14 ^^ 













5 6 7 8 

Values of Constant K 



10 



U 



12 



13 



CHART 6. RELATIONSHIP BETWEEN CIRCULAR PITCH AND FACTOR K USED IN ESTIMATING 

WEIGHTS OE SPUR GEAR BLANKS. 



112 AMERICAN MACHINIST GEAR BOOK 

In endeavoring to apply this table to practice it was found that the square 
of the pitch (C^) was not a correct factor, and a separate table was made up 
for it. 

After repeated trials and corrections in the value of the constant for the 
number of teeth, it was noticed (when close results were finally obtained) that 
the values ran parallel with the number of teeth, and were, therefore, dropped, 
changing the value of the constants to the square of the pitch, so called, to 
obtain like results. 

To find the weight, therefore, it is only necessary to find the combined 
product of the number of teeth, face, and a constant given for the pitch: 

Weight = number of teeth y. F X K. 

The accompanying Chart 6 gives the value of K. 

This formula has its limitation; it cannot be used for low numbers of teeth, 
varying with the pitch, also for large gears there is a variation in design that 
cannot well be covered by one constant. However, the proper constant may 
be readily determined by trial for different constructions. In general it must 
be used with discretion, but is invaluable as a check. The finished weight is 
found by deducting 3 per cent. 

The following equation seems to give a close approximation to the curve 
of Chart 6 throughout the ordinary working range up to 3^^ inches circular 
pitch: p'^ = i.^SK, in which p' is the circular pitch and K the constant 

p'^ 
previously referred to. Transposing we have K = — ^, or substituting in 

equation for weight of spur gears 

Weight = -^--^ X number of teeth X face. 

Beyond 3 3-^ inches circular pitch there is considerable variation between 
the curves from the actual data and from the equation. However, as stated, 
the great variations in the design of large gears cannot be cared for by a 
single constant. In using the formula its limitations should be carefully 
understood. As a matter of interest, it might be stated that for 6 inches 
circular pitch the curves are again practically in agreement. 

ARMS FOR SPUR GEARS* 

In deducing a formula for gear arms it is assumed that the thickness of 
the rim is sufficient to distribute the load between the arms; this assumption 
is quite justified, as such a depth is necessary to prevent bending of the rim 
between adjacent arms. By equating the expressions for the tooth strength 
and that of a beam supported at one end and loaded at the other, the general 
expression arrived at is 



* 



Henry Hess. 



when 



GEAR PROPORTIONS 1 13 

p'^RiN — 7) 
Z = — 7 — - — for circular pitch, and 

Z = ^-, — - — for diametral pitch, 

Z = modulus resistance of arm cross-section. 
p' = circular pitch. 
p = diametral pitch. 

F 

R = ratio of face width to circular pitch = —,- 

P 
F = face width. 

N = number of gear teeth. 

A = number of arms. 

If it is preferred to use the face width itself, instead of its ratio to the 
circular pitch, then 

Z = ; — for circular pitch. 

50.4 

Z = r-r^ for diametral pitch. 

By these formulas the dimensions of a gear arm of any section whatever 
can be determined. 

As by far the great majority of cast gear arms are of elliptical cross-section, 
these expressions are reduced by inserting the terms of the modulus of re- 
sistance of an ellipse in which the major axis is double the minor, and the 
formulas become, when E = the thickness of the arm at its base, and 

2E = the wudth of the arm at its base; 



E = J(^)^ = /^';^y^^[ for circular pitch; 



E = {li^-4^ = {j^-^^^l for diameteral pitch. 

To reduce the labor involved by the mathematical solution. Charts 7 and 8 
have been constructed, one for diametral pitches ranging from 10 to 3, and 
the other for circular pitches ranging from i to 3 inches; as 3 diametral pitch 
is very nearly equal to i-inch circular pitch, the second chart extends the 
range of the first without a break, so that, between the two, any case Ukely 
to arise will be taken care of. Diametral pitch is given in Chart 7, as the 
more general practice uses that for small and medium-sized gears, while for large 
work; circular pitch is generally employed, and is therefore used as the basis of 
Chart 8. Two charts are required, as the inchnation of the pitch diagonals 
toward the end values would become too slight to admit of accurate reading 
on a single one. 

By tracing the number of teeth from the bottom scale to the pitch diagonal 



114 



AMERICAN MACHINIST GEAR BOOK 




Directions: Trace up from the Number of Teeth to the Pitch Diagonal, then Horizon- 
tally to the Vertical above 300 Teeth, then Parallel with the Nearest Diagonal to the Vertical 
Headed with the Arm Number E; then Trace to the left Horizontally to the Vertical R repre- 
senting the Ratio of Face Width to Circular Pitch R. Take the Nearest Diagonal as the 
Arm Base Thickness E. 

CHART 7. PROPORTIONS OF GEAR ARMS FROM DIAMETRAL PITCH. 



GEAR PROPORTIONS 



ITS 




M ^. 



Directions: Trace up from the Number of Teeth to the Pitch Diagonal, then Hori- 
zontally to the Vertical above 300 Teeth, and Trace Parallel to the Nearest Diagonal to the 
Vertical Headed with Number of Arms: then Trace to the left Horizontally to the Vertical 
R representing the Ratio of Face Width to Circular Pitch. Take Nearest Diagonal as 
Base Thickness of Arm E. 



CHART 8. PROPORTIONS OF GEAR ARMS FROM CIRCULAR PITCH. 



Ii6 AMERICAN MACHINIST GEAR BOOK 

in the main portion of the circular pitch chart and referring the intersection 
to the vertical scale under 8, the value is found of 

(N - fp') f^^ ^ ^ g ^j.j^g^ 
20 A 

For any other number of arms this is modified by employing the auxiliary 
portion of the chart at the right, referring the value just found along or be* 
tween the nearest slant lines to intersection with that vertical representing 
the number of arms actually used. By now tracing this hight horizontally 
to the left to that vertical R representing the particular ratio of face width 
to circular pitch employed, and taking a reading from the nearest slant line 
crossing this vertical, the value first found is multiplied by R and the cube 
root extracted. 

Past practice gives a face width between two and three times the circular 
pitch, but as the tendency is now toward a wider face, ratios from i3^^ to 4 
are given. 

Concise directions are printed with the charts. Dotted trace lines of the 
following examples are also drawn in: 

Example i. Given a gear of 100 teeth, 4 diametral pitch, 6 arms and ratio 
of face width to circular pitch = 23^^. 

Trace on Chart 7 100 teeth up to diagonal for 4 pitch, horizontally to num- 
ber of arms 8, slantwise up to number of arms 6, horizontally to the left to 
ratio 2}^, which is intersected between ^^q inch and i inch; therefore thick- 
ness of arm is to be taken as i inch and width as 2 inches. By calculation the 
dimensions are 0.96 inch and i.gS inches. 

Example 2. Given a gear of 270 teeth, 2-inch circular pitch, 6 arms and 
ratio of face width to circular pitch = 2. 

Trace on Chart 8 as before and find 33^:^ inches full as thickness of gear 
arm at base, and 6j^ inches full as width. The calculated dimensions 3.27 
inches and 6.54 inches agree almost absolutely with the much more quickly 
obtained values by the chart. 

In large arms the designer will frequently prefer a cored section. A sat- 
isfactory one will be that of Fig. 77, in which major and minor axes of both 
core and arm are relatively as 2 to i. By equating the moduli of resistance 
for solid and hollow elliptical sections of these proportions, it is found that 

E^ = jz , in which E is the thickness of the solid arm as obtained by 

chart or formula; d and D are dimensions of the cored arm. See Figs. 77 
and 78. 

In order to lessen the work of making the core box by substituting flat sur- 
faces for curved ones, an approximation like Fig. 78 will add but slightly to 
the weight, as is shown by the ellipse dotted in for comparison. 

The ellipse outlines are formed of circular arcs struck from four centers, 
which will approximate very closely to the true ellipse. The construction 
of the core sides is readily apparent from the sketch. 



GEAR PROPORTIONS 



117 



The arm taper is stated as i in 32 and 16, respectively, for the arm thick- 
ness and width; this gives a pleasing appearance for a moderately long arm, 




FIG. 77. PROPORTIONS OF HOLLOW ARMS. FIG. 78. 



but it is not a hard-and-fast rule, as a greater or lesser taper may be employed 
to suit the designer's fancy without affecting the strength of the arm, unless 
the taper is made so excessive as to bring the dimensions at the rim down to 
one-half of these at the base. 

As the tooth and arm are of the same material, the method is satisfactory 
for all cast gears, but this must not be interpreted to mean that this or any 
other formula will prevent shrinkage , , 

strain due to relatively large hubs or very 
heavy rims; where these occur, great care 
must be exercised in the foundry, and 
it will also not be amiss to add a generous 
amount of metal to the arms. 

RIM GEAR PROPORTIONS 

Where steel castings prove inefficient 
for the work intended, forged steel 
rims, designed somewhat as illustra- 
ted in Fig. 79, are used. The center is 
made of cast steel of a heavy pattern, 
the forged rims being shrunk thereon, 
obtained of a higher grade of steel than is possible in the casting, and is 
free from hidden flaws, also as the rim is renewable, it is a much better and, 
in the end, a cheaper proposition. The rims are made of a forged billet, 
which is first pierced and then rolled into shape by the same process em- 
ployed for locomotive tires. 

There are many variations of this design, but it is thought best 




FIG. 79 

As this rim 



PROPORTIONS OF RIM GEARS. 



material may be 



ii8 



AMERICAN MACmNTST GEAR BOOK 



to make the face of the center narrov/er than the rim on account of possible 
unevenness in fitting, and turn a shoulder on center to bring rim up true, 
instead of depending upon parallels or surface plates. Also it will be desir- 
able in many cases to replace rim without removing gear from the shaft. 
This is accomplished by rapidly heating rim by means of a circular gas or 
oil burner made to suit diameter of gear and protected by asbestos cover to 
localize the heat. 

DESIGN OP BEVEL GEARS 



The rules for the design of spur gears may be applied to helical, herring- 
bone, worm, and spiral gears, using the circular pitch as a basis. Also for 
bevel gears, taking the proportions from the large end of the tooth. The 

average design of bevel gears is 
shown in Fig. 80. The hub should be 
carried well back of the face and 
connected to the rim by ribs, 4, 6, 8, 
or 10 in number, depending upon the 
diameter of the gear. The hub 
should not be carried too far in the 
front, or small end, of the gear, as 
a long hub will make it impossible 





FIG. 80. BEVEL GEAR PROPORTIONS 



no. 81. BEVEL GEAR WITH LONG FRONT 
HUB EXTENSION. 



in many cases to cut the teeth, for if this hub is carried too far it will inter- 
fere with the operation of the machine. See Fig. 81. A small hub, how- 
ever, should always be put on the front end of the gear, otherwise it will be 
necessary to counterbore to secure a finished bearing. 



RAWHIDE GEARS 



Rawhide gears are commonly made with brass flanges on either side of the 
face to hold the rawhide in position and to engage the mating gear, unless 
rawhide contact alone is desired. It is practice to speak of the face of the 



GEAR PROPORTIONS 



119 



rawhide gear as including these flanges. See Fig. 82. There is no great gain 
in preventing the flanges from coming into contact, and to make a rawhide 
gear without cutting through the flanges, as per Fig. S$, is unnecessary and 
expensive. 

There is no reason why steel flanges cannot be used in place of brass, es- 
pecially for the larger sizes; boiler plate will be found excellent for this. The 
thickness of the flange is something that varies greatly, although ^q oi the 
circular pitch is a fair average. 



^ 



. >'i|i|iii|i|ll'i'Xi!ii"liii'!'i'i>i'i'l'i"i 



Rivet 



i 



;|i|}|l'!!!'ll;i!ii!i'H!!lli'!l!!!llt ill 



Keyseat 



m 



i 



g 



^WF 



' Rawhide '1 In' '1 



I I 

— H h Rawhide ^ ]<- 

! I ! I 



f< ^Face- 



FIG. 82. RAWHIDE GEAR. 



^^ 



I I 

[< ^Engaging-Gear ^ 

,1 0.0625 pi I, 

-^k— for Clearanc?5t(<- 



^'iii'i!i!lMii!iiii'ii!il!il'||i!i!i!il|ili'ili 




Rivet 




Keyseat 



'Ili^^S S I'll! 

'l'i''ili'JihlllilllMl|ll|l 



0.3125 p' I 

h Face >r^ k— 

I ! ! I 
H^ ijength-over-all H 

I ' 

FIG, 83. SHROUDED RAWHIDE GEAR. 



For the larger rawhide gears it is recommended that bolts instead of rivets 
be used, as it is impossible to otherwise draw up a wide-face rawhide gear. 
The bolt head may be countersunk so that one side of the gear will be flush. 
It is also sometimes possible to put the nut in a counterbore; this depends, of 
course, on the design of gear. 

Rawhide gears to run loose on the shaft should be bushed. When it is 
necessary to move a rawhide gear on a spline one of the flanges should be made 
as part of the bush, as per Fig. 84. The usual design of the larger size rawhide 
gear is shown in Fig. 85. 



I20 



AMERICAN MACHINIST GEAR BOOK 



Rawhide bevel gears are designed similar to Fig. 86. Both ends of the 
teeth must be flanged to facilitate the cutting of the teeth. 

Fiber is often used in place of rawhide, but is usually more brittle and 
has a tendency to wear the engaging gear, although made of a harder material. 




FIG. 84. BUSHING AND FLANGE IN ONE 
PIECE. 



^ 








iiiiii||iiii|i|iii!|ii!ii;i|ii|i||i 
^iiiiiiiiiii|i|ii|iliiiii|ii 



Jim 



war 



l|l|i|iUl>|i|i|!|i|i|i,l|l,i|i| 



m 



V//////////////////////////////////// 



\ 



/ 



^ 



7mmm777mm77P7777?77^p^^m 




iil'iii|ill'|i|l|l |l|i 
llRaKvihideli 1" 
iiiriTihr l["l'|llilll'l|ll|l,i 

|!|iiii!i||l!!!'!ilili|lil!l|ii|'i|ill|i| 



il||l|l 



Pace- 



FIG. 85. DESIGN OF RAWHIDE GEAR. 



Fiber gears are ordinarily furnished without flanges. Gears are also made of 
laminations of rawhide and bronze, or fiber and bronze, but not to any great 
extent. 

A fiber stress of 5 ,000 pounds per square inch is amply safe for calculating 
the strength of rawhide gears. 



GEAR PROPORTIONS 



121 



MORTISE GEARS 

The mortise gear is composed of a cast-iron rim, containing cored slots, 
into which wooden teeth are driven. See Fig. 87. These gears cannot 




riG. 86. RAWHIDE BEVEL GEAR. 

compare either in efficiency or cost to a properly cut cast-iron gear, but are 
still used in many places where excessive noise is prohibitive. The wooden 




Tooth 



TwoTCeys, One to 

Drive from each 

Side of Face, 



FIG. 87. PROPORTIONS OF MORTISE GEAR BASED ON 
I -INCH CIRCULAR PITCH. 



122 



AMERICAN MACHINIST GEAR BOOK 



teeth are made either of apple or maple, treated in linseed oil and cut to the 
proper form after being inserted in the cored rim; otherwise the spacing of 
the teeth would be governed by the spacing of the cores, which can never be 
very accurate. Replaced teeth must be fitted and shaped by hand. The 

strength of the mortise gear is governed by the 
thickness of the iron teeth in the engaging gear, 
which are made 0.35 of the pitch instead of 0.5 
pitch, as for cut gears. The outside diameter of 
the cored rim is turned to the dedendum diameter 
that the teeth will be set an equal distance from 
the center. 

KEYSEATS 

The commonly accepted keyseat standard 
is that of Jones and Laughlin, as per Table 
17. In this the width of the key approximates one-quarter the shaft 
diameter. A square key is used, one-half being in the gear and one-half 
in the shaft. Dimensions for taper keys are included in this table. A taper 
keyseat, however, is nothing more than a means of tightening up a poor fit; 
a gear properly fitted to the shaft will not work loose if a straight key is used 
with clearance on the top. The best fit can be spoiled by a little carelessness 
in driving a taper key, as it is sure to make the gear run out if driven 








KEYSEAT TAPER 








BORE 


}4 INCH PER FOOT 


STRAIGHT 


KEYSEAT 


















SMALLEST 










WIDTH 




WIDTH 


HIGHT 






**A" 


HIGHT 


A 


"b" 




IN. IN. 


IN. 


IN. 


IN. 


■IN. 




8 iofA 


2 


\\ 


2 


I 




M " iV^ 


iVs 


Vs 


1% 


% 




7M " 6J^ 


iM 


33 


iH 


Vs 




6M " m 


iVs 


H 


iVs 


% 




6H " 5^ 


tVi 


y^ 


1V2 


M 




sH " sVs 


1% 


2.9 

64 


iVs 


% 




sH " 4^ 


^y4. 


\\ 


iM 


Vs 




4H " 4^6 


iVs 


Vs 


iVs 


% 




4^ " 4^6 


lYe 


hk 


lYe 


^^ 




4^ " 3% 


I 


H 


I 


1/2 




sVs " 3% 


!¥6 


^6 


% 


Ifi. 
3 2 




sVs " 3^6 


Vs 


H 


Vs 


Ji6 




sVs " 3^6 


% 


-a 


% 


il 




sVs " 2% 


% 


M 


H 


Vs 




2ys " 2% 


% 


15. 

64 


% 


U 




2% '' 2j{^ 


% 


il 


Vs 


Ve 




2% " 2Xe 


%, 


5i6 


% 


9 
'32 




23^ " 1% 


Yi 


H 


¥1 


M 




iVs " 1% 


% 


9 

'6 4 


Ji6 


f, 




IVS " 1^6 


^ 


% 


Vs 


%, 




IVS " 1^6 


^6 


i-i 


V, 


^h 




iVs " I 


M 


^h 


34 


Vs 



Table 17 — Standard Keyseats 



GEAR PROPORTIONS 



123 



the least bit too tight. The taper key is applicable to only the heaviest 
work where the mass of metal will prevent any such distortion. 

For gears that are to be hardened it is important that there be a fillet in 
the corners, the top of the key being beveled off to suit, otherwise a crack is 




FILLETED KEYWAY. 



very liable to start from the sharp corner of the keyway. For that matter, 
this would be an excellent plan to adopt for all keyways. See Fig. 88. 
For machine tools and automobiles the Woodruff key is generally used. 



-a— 




I /' 



"1 
. I 
\ I 












CENTER OF 










CENTER OF 




DIAM- 


THICK- 


DEPTH 


STOCK, FROM 




DIAM- 


THICK- 


DEPTH 


STOCK FROM 


NO. OF 


ETER 


NESS 


OF 


WHICH KEY 


NO. OF 


ETER 


NESS 


OF 


WHICH KEY 


KEY 


OF 


OF 


KEY- 


IS MADE, TO 


KEY 


OF 


OF 


KEY- 


IS MADE TO 




KEY 


KEY 


WAY 


TOP OF KEY 




KEY 


KEY 


WAY 


TOP OF KEY 




a 


h 


C 


d 




a 


h 


C 


d 


I 


'A 


Xe, 




-h 


B 


I 


Yfi 


"32" 


Ye 


2 


3^ 


-3.. 


/4- 


-h 


16 


1% 


¥6 


^ 


-6\ 


3 


M 


Vs 


X. 


-h 


17 


iVs 


3^2- 


i-^ 


-6^- 


4 


Vs 


■h 


-^- 


^6 


18 


iVs 


Va 


Vs 


-h 


5 


H 


% 


X, 


X, 


C 


i^ 


X, 


_5.. 
32 


-h 


6 


Vs 


:tl 


-h 


X, 


19 


^H 


Ye 


liV 


-ti 


7 


Va 


Vs 


Yfi 


X, 


20 


^Ya 


3V 


-.h 


-6\ 


8 


M 


3% 


^4- 


X, 


21 


iM 


Va 


Vh 


i^ 


9 


% 


% 


.a. 

3 2 


X, 


D 


iM 


X, 


-h 


-h 


10 


% 




ft 


X, 


E 


iM 


% 


Ye 


-h 


II 


Vs 


'%, 


3^2- 


X, 


22 


i^ 


Va 


H 


■h 


12 


% 


.1. 
3 2 


-h 


X, 


23 


i^ 


Ye 




3% 


A 


Vz 


M 


% 


Ye 


F 


i^ 


% 


Ye 


3% 


13 


I 


^6 


-h 


X, 


24 


i3^ 


Va 


Vs 


eV 


14 


I 


-1- 

3 2 


-h 


Ye 


25 


^¥2 


Y« 


^h 


/^ 


^5 


I 


M 


3/8 


Ye 


G 


1X2 


% 


Ye 


64 



Table 18 — Woodruff Standard Keys 



124 



AMERICAN MACHINIST GEAR BOOK 





13 o 



26 
27 
28 
29 
30 







o>; 


H W 


sS 


K ^ 




y ph 


Pi w 


i-i c 

Q 


a 

H 




a 


5 


C 


2% 


% 


_3_ 
3 2 


2^ 


1^ 


1/^ 


2l^ 


^6 


•> 
32 


2^ 


^/^ 


^6 


3^ 


^8 


% 



t. § bH O >< 

rt fH ^ W , 
U H ^ t/3 g 



17. 
3 2 
il 
3 2 
11 
3 2 
11 
3 2 

/16 



P4 



Pi >^ 




72 


05 


_ H 


S w 


H W 


<; 


w < 


s « 


w w 


w ^ 


H 1-1 


s 


g 


H >H 


Q fH 


b fH 


< ^ 


^ & 


Pi w 


^ 


^ 


s^ 


a 


w 2 


e 




a 


6 


c 


3% 


31 


33^ 


JTe 


3^2- 


li^ 


32 


3.4 


•K2 


34 


;^f 


?>2> 


3,^ 


% 


9 

3 2 


Ti^/ 


34 


3M 


^8 


% 


^6 











p^ ^ ^ 2 ^, 

g o w H a 

H M g < O 

^ S< ^ 1^ o 

H O ►" (^ 

^ [^ ^ ^ g 






^6 



Table 19 — Woodruff Special Keys 



DIAMETER 
OF SHAFT 


NUMBER OF 
KEYS 


\ DIAMETER 
OF SHAFT 


NUMBER OF 
KEYS 


DIAMETER 
OF SHAFT 


NUMBER OF 
KEYS 


%-Vs 
%-H 

% 


I 
2,4 

3, 5 
3^ 5, 7 
6, 8 


1^6-1)^ 
1^6 

iM-i^6 


6, 8, 10 
9, II, 13 

9, II, 13, 16 

11, 13, 16 

12, 14, 17, 20 


I^-lJi^ 
iM-I^ 

l%-2 

2%r2y2 


14, 17, 20 

15, 18, 21, 24 
18, 21, .24 
21, 25 

2S 



Sides of Key a ueat Fit 
Top of Key Tapers M Inch per Foot 



Woodruff Standard Keys 10 Use with Various Diameter Shafts. 

For heavy work the ordinary key will not answer : it is often necessary to 
put in two keys diametrically opposite, and for extremely heavy work, what 

is known as the "Kennedy" key, Fig. 89, 
is used. This is the only key that will 
answer the requirements of rolling-mill 
work. At the armor-plate mill of the 
Carnegie plant this type of key is used in 
the 22-inch shaft of rolls that reverse on 
an average of 20 times per minute. No 
other key would stand up to this work. 
These keys are made approximatly one- 
quarter of the shaft diameter, and lo- 
cated in the gear so that the corners of 
the keys intersect the bore. It is not 
necessary for the bottoms of the keys to 
be on a vertical line. The keys are made 
to a taper of J^ inch per foot on the 




fig. 89. the KENNEDY KEY. 



GEAR PROPORTIONS 



125 



top for a driving fit, the sides being just a neat fit. The shaft is first bored 
for a press fit, then rebored about J ^4 of the shaft diameter off center; the 
keyways are cut in the eccentric side. That portion of the bore opposite 
the keys remains as originally bored to within one-tenth of the shaft 
diameter below the center line. 

The ''Kennedy" key is especially desirable where it is necessary to move 
the gear for any considerable distance on the shaft before securing. 

For street-railway work the gear is often pressed on the axle, not using a 
key of any description. 

Where a sliding gear is used for heavy work, three keys, having radial sides 
as illustrated by Fig. 90, are generally employed. An example of this is the 





FIG. 90. KEY^VAYS FOR HEA\^ SLEDIXG GEAR. FIG. QI. FOUR-KFYED SLIDING GEAR. 

gear drive for vertical rolls. This style of key has a distin^.t advantage over 
the four-key type with parallel sides as illustrated by Fig. 91. 

BORE 

The bore of a gear is supposed to be standard, any allowance for a fit being 
made in the shaft. This follows out the practice of the manufacturers of 
cold-rolled shafting, that is, to make the shaft enough under size for a sliding 
fit in a gear or pulley which is bored standard. 

An exception to this is when press or shrink fits are desired. In this case 
the allowance is made in the gear, the shaft being turned standard. 



PRESSURES AND ALLOWANCES FOR FORCE FITS 

The Lane & Bodley Company of Cincinnati, O., furnishes the following 
data. For several years this firm has been keeping a record of observations 
on press fits with a view to making an analysis of them when a sufficient body 
of data had been accumulated, and thus obtaining a guide for future practice. 
Hundreds of cases of such press fits have been recorded, forming a body of 
data which is probably unequaled. See Chart 9. 



126 



AMERICAN MACHINIST GEAR BOOK 




20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 210 220 230 240 250 260 270 280 290 

Pounds 

CHART 9. FORCE-FIT RELATIONS SHOWING LANE & BODLEY PRACTICE, 



GEAR PROPORTIONS 127 

MEASUREMENTS AND PRESSURE READINGS 

In these records the measurements have been made with great thorough- 
ness. Both plug and hole have been measured on two diameters and at both 
ends, the average of these micrometer readings being taken as the true 
diameter. The pressures have been read at the beginning, middle, and end of 
the length of the fit; the material of both plug and ring, the length of the fit, 
the radial thickness of the hub, the areas and volumes of the fitted surfaces, and 
some other minor points have been noted, 24 entries being regularly made for 
each case. The resulting chart will thus be seen to have a very broad 
foundation. 

In ordinary cases the quantities which are fixed by the conditions are the 
nominal diameter and length of the fit, the radial thickness of the hub, and 
the material. With these given it is required to find the press allowance for a 
given pressure to force the plug home. 

HOW THE PRESSURE VARIES 

Regarding the influence of these various factors, the pressure varies: 

1. Directly as the surface of the fit for a given diameter. 

2. Directly as the press-fit allowance, this allowance being such as not 
to stretch the metal beyond the elastic limit. 

3. As some function of the radial thickness of the hub, which, while not 
determined mathematically, is shown in the chart. 

4. With the materials used in a manner not yet determined owing to in- 
sufficient data. The chart is for steel or iron shafts and cast-iron cranks. 
Cast-steel cranks require much heavier pressures for the same press-fit al- 
lowances, but how much heavier cannot at present be said. 

5. With other conditions, which cannot be formulated, and which lead to 
erratic results in the observations make it impossible to formulate a rule or 
construct a chart which shall give other than approximate results. 

VARYING CONDITIONS 

Among these are the nature of the surfaces as regards smoothness, the 
varying character of materials going under the same name, the shape of the 
crank and the speed with which the work is done. The term cast iron 
includes materials of widely varying hardness and other properties, and it is 
apparent that the web of a disk crank would have an influence not expressed 
by the radius of the hub. If a counterbalance were cast in the disk, this and 
the crank arm would naturally produce an effect on the effective thickness of 
the hub which would be different from the effect of the arm alone on a plain 
crank. Again, with a plain crank, the arm, being taper, would reinforce a 
greater arc of the pin eye than of the shaft eye. 



128 AMERICAN MACHINIST GEAR BOOK 

STILL ANOTHER FACTOR 

Another factor, which no doubt introduces some of the discrepancies of 
the diagram, is that while most of the shafts were of steel, some were of 
wrought iron, and no discrimination between these materials has been made 
in the analysis. These considerations explain the erratic results obtained, 
but it is nevertheless plain that the observations follow the general direction 
of the curves in a very marked manner. 

AVERAGES SHOWN BY THE DIAGRAM 

The plotted observations, it should be remarked, are in most cases the 
averages of many. The figures attached give the percentage of radial hub 
thickness to plug diameter. It will be observed in this connection that the 
discrepancies grow less as the diameters increase. This is doubtless chiefly 
due to the fact that the percentage of error is always greater with small 
experiments than with large. Its effect is to give increased value to the 
diagram when used with large sizes where it is most needed. 

THE PROBLEM SOLVED BY THE CURVES 

Of course, the holding power of these fits is the real thing desired, but it is 
obvious that the probabiUty of adequate experiments being undertaken to 
determine this in large sizes is sHght. The problem as it presents itself in 
the shop calls for the determination of the press-fit allowance to give a 
required pressure in forcing the parts home, and this the present diagram 
solves with a degree of accuracy sufficient for most purposes. 

In order to reduce the size of the diagram, the portion applying to diam- 
eters above 13 inches has been detached and placed at the right. For these 
large sizes the observations are too few to justify the drawing of more than 
one curve. The lubricant used in all cases was linseed oil. 

DIRECTIONS FOR USE 

Select the curve which gives the ratio of the radial thickness of the hub 
divided by the diameter of the plug. Below the point of intersection of the 
plug diameter line with the selected curve, read pounds. Multiply this read- 
ing by the area of the fitted surface in square inches, and by the number 
of thousandths of an inch allowed for the press fit. The result will be the 
pressure in pounds carried to force the plug home. 

AN EXAMPLE 

The following example illustrates the use of the diagram: Diameter of 
plug, 8 inches, length of fit, 6 inches, diameter of hub, 16 inches, press-fit 



GEAR PROPORTIONS 129 

allowance, 0.020 inch. Required, the pressure to force the parts together. 

Radial thickness of hub = 4 inches _ 
Diameter of plug = 8 inches 

Finding a point on the 50 per cent, curve opposite 8 inches, and tracing 
downward, we find 52 pounds. Area of fitted surface = 8X3.1416X6 = 
150 square inches, 52 X 150 X 20 = 156,000 pounds = 78 tons. 



SECTION V 

Bevel Gears * 

Cut bevel gears may be divided into three classes: First, that in which the 
teeth are of uniform depth throughout their entire length and of exact 
profile; second, that in which the depth of teeth correctly decreases toward 
the apex, but in which the teeth are originally cut with a correct profile at 
one point only, necessitating subsequent work on the teeth; and third, that 
in which the true taper type of teeth with correct profile at all points is gen- 
erated by a process reproducing the rolling action of a pair of gears in mesh. 
These three classes may then be referred to according to the form of tooth 

cut: First, a modification of the correct 
tooth; second, an approximation of the 
correct tooth; and third, the correct form 
of tooth. 

The principal relationships of the 
various dimensions, angles, etc., of stand- 
ard bevel gears of true form, in accord- 
ance with the notations and formulas on 
the succeeding pages, are as follows: 

For gears meshing at right angles, 
the tangent of the center angle of a bevel 
gear is found by dividing the number of 
teeth in the gear by the number of teeth 
in the pinion; obviously the complement of the center angle of the gear is 
the center angle of the pinion. If the axes of the gears do not meet at right 
angles more complicated calculations are necessary. See the succeeding 
formulas. 

The face angle is found by adding the angle increment, the angle between 
the pitch line and the face of the tooth, to the center angle. 

The back angle is the complement of the center angle. See Fig. 92. 
The cutting angle is found by subtracting the angle decrement, the angle 
between the pitch line and the bottom plane of the tooth, from the center 
angle. 

When bevel gears are to be milled with parallel depth teeth, the back 

angle becomes the complement of the face angle, and the cutting angle is 

obtained by subtracting the angle increment instead of the angle decrement 

from the face angle, making the face angle of the gear the same as the cutting 

* See Section XVI for Rolled Bevel Gears and Section XIV for Special Bevel Gears, 

130 




TIG. 92.. 



-90"^ 
Face Angle Back Angle 

LOCATION OF FA.CE AH.D BACX 
ANGLES. 



BEVEL GEARS 



131 



angle of the pinion and making the clearance the same at both ends of the 
teeth. The back and center angles are the same. 




FIG. 93. DIAGRAM AND NOTATION FOR BEVEL GEAR. 



The tangent of the angle decrement is found by dividing twice the sine of 
the center angle by the number of teeth. This holds true, however, only 

for gears having an addendum of . or 0.3183/?' (Brown & Sharpe standard). 



132 



AMERICAN MACHINIST GEAR BOOK 



When special teeth are used, the angle increment equals the addendum 
divided by the apex distance ( )' and the angle decrement equals the adden- 
dum plus the clearance divided by the apex distance ( Y When the 

addendum equals ~ ? the angle increment is found by dividing the product 

of 2.314 and the sine of the center angle by the number of teeth. 

The diameter increment is found by multiplying the addendum by the 
cosine of the center angle. Twice this diameter increment added to the 
pitch diameter of a bevel gear gives its outside diameter. 




riG. 94. DEPTH OF FRONT OF RIM. 



The apex distance is found by dividing the pitch diameter by twice the 
sine of the center angle and is the same for both gears of a pair. 

When turning gears in large quantities, the work can be carried forward 
more expeditiously if the length of face is given measured parallel with the 
center line of the bore. This distance R is found by multiplying the face by 
the cosine of the face angle. 

Time may also be saved in turning bevel gears if the depth of rim at the 
front end is given. This will allow the workman to finish the face, front end, 
and bore of the gear during the first operation, while it is held in the chuck 
by the back hub. Ordinarily the depth is made 0.2 inch per inch circular 
pitch, but this can be varied to suit requirements, provided the rim depth 
is sufficient to allow for a full tooth at the small end. See Fig. 94. 

The pitch diameter at the small end of the teeth is found by subtracting 



BEVEL GEARS 133 

twice the product of the face and the sine of the center angle from the pitch 
diameter at the large end of the gear. 

The outside diameter at the small end bears a similar relationship to the 
outside diameter at the large end. 

The number of teeth for which the cutter should be selected is equal to 
the number of teeth in the gear or pinion divided by the cosine of its center 
angle. 

The thickness of the tooth at the small end is found by dividing the prod- 
uct of the thickness of the tooth at the large end and the difference between 
the apex distance and the face by the apex distance. Or: 

a:t::a-b:t'. Therefore t' = ^i^-1^- 

a 

All dimensions for the small end of the tooth may be determined in this 
proportion. 

NOTATION AND FORMULAS FOR MITRE GEARS 

Center angle. £ = 45 degrees. 

Face angle. F = 45 deg. + J- 

Cutting angle, C = 45 deg. — K. 

Angle increment. Tan. J = ) or T^ when s = -• 

^ a N p 

Angle decrement. Tan. K = -y or '.^ when s = . 

^ a N p 

Diameter increment. F = 50.70711. 

Backing. 7 = 50.70711. 

Outside diameter. D = d -\- 2V. 

Apex distance. a = j or d 0.7071 1. 

0.1414 

Distance from apex to point of tooth — 

large end. Parallel with axis. F = F. 

2 

Face, measured parallel with axis. R = b cos. F. 

Depth of rim at small end. M = ^-^ 0.7071 1. 

a ' 

Pitch diameter at small end. d^' = d — 2 {b 0.707 11). 

Outside diameter at small end. L>^ = D — 2 {b sin. F). 

Number of teeth for which cutter 



should be selected. = 

Thickness of tooth at small end. t' = 



N 



0.70711 

/ (a - b) 
a 



134 



AMERICAN MACHINIST GEAR BOOK 




FIG. 95. MITRE GEAR. 



NOTATION FOR BEVEL GEARS 



AXIS AT 90 DEGREES 

E = center angle. 
F = face angle. 
C = cutting angle. 
/ = angle increment. 
K = angle decrement. 

V = diameter increment. 

Y = backing, or distance from point of tooth to pitch line. 
d = pitch diameter. 

D = outside diameter. 
p' = circular pitch. 

p = diametral pitch. 
d^ = pitch diameter at small end. 
D^ = outside diameter at small end. 

t = thickness of tooth at largest pitch diameter. 

t' = thickness of tooth at smallest pitch diameter. 

a = apex distance. 
H = distance from pitch line to apex. 
F = distance from point of tooth to axis of mating gear. 
R = face measured parallel with axis. 
M — depth of rim at front end. 

h = face. 

5 = addendum. 
W = whole depth. 
N = number of teeth. 



FORMULAS FOR BEVEL GEARS AT ANY ANGLE 

Many formulas have heretofore been pubhshed for determining the angles 
and dimensions of bevel gears not at right angles, all of which are more or less 
confusing. It is simply a matter of finding the center angles; all other angles 



BEVEL GEARS 



135 



and dimensions are obtained as for ordinary bevel gears at right angles, each 
gear being figured separately. This also applies to the use of tables, the table 



GEAR 


PINION 


REMARKS 


REQ. 


FORMULA 


REQ. 


FORMULA 


E, 


Tan. E2 = -j^— 


E, 


90° - El 




F. 


E, + J 


F, 


Ei-\- J 


Subtract from 90* 
for use in lathe. 


C. 


E^ -K 


Ci 


El -K 




J 


2 sine E-i 
Tan. J- ^^ 


/ 


2 sine El 
Tan. J - ^^ 


Same for both gear 
and pinion. 


K 


2.314 5z«e £2 
Tan. K - ^^ 


K 


2.^14 sine El 
Tan.K- ^ \^ 


Same for both gear 
and pinion. 


V, 


S COS. E2 


F, 


S COS. El 


Vi or pinion same 
as F2 for gear. 


Y, 


s sine Ei 


Y, 


S sine Ei 


Yi for pinion same 
as F2 for gear. 


D, 


d-x + 2F2 


D, 


d. + 2F1 




a 


di 


a 


di 


Same per both 
gear and pinion. 




2 sine E-i 


2 sifie El 


P. 


^-> 


P. 


~-r. 




R. 


b COS. F-i 


i?i 


b cos. Fi 




H, 


d, 

2 


Hy 


d, 
2 




M-2 


W'+o.2p' {a-h) 

sine El 


Ml 


W + 0.2 p' {a-b) , 

sine E\ 

Of 




d," 


d-i ~ 2 (b sine E-i) 


d,^ 


di — 2 (b sine E\) 




D,- 


Di — 2 (b sine F2) 


Di^ 


Di - 2 (b sine Fi) 




Cutters 


N2 

COS. Ea 


Cutteri 


Ni 
COS. El 




/' 


t(a -b) 
a 






Same for both gear 
and pinion. 



Formulas for Bevel Gears, Shafts at 90 Degrees 



being entered for each gear separately, according to its center angle and in- 
dependent of its mate. 

When the axes are not at right angles there are four other combinations: 
Axes less than 90 degrees, see Fig. 96; axes greater than 90 degrees, Fig. 97; 



136 



AMERICAN MACHINIST GEAR BOOK 



crown bevel gears, Fig. 98; and internal bevel gears, Fig. 99. The center 
angles for these gears are found as follows: 

A^2 = number of teeth in gear. 
Ni = number of teeth in pinion. 




FIG. 96. BEVEL GEARS WITH AXES LESS Am^an mMnUt 

THAN 90 DEGREES. j.jG gy^ ^^.^j^ ^^^^^ ^^^ ^^^ GREATEK 

THAN 90 DEGREES. 



Tan. E = 



sin. B 



N, 



Tan. E = 



N2 
E' = B - E 



+ COS. B 



sin. (180 — B) 

-^ COS. (180 — B) 

i\ 2 



E' = B - E. 




Avicrican JiTaeftiTait 
FIG. 98. CROWN BEVEL GEARS. 



E = 90°. 

E' = B - 90. 



FIG. 99. INTERNAL BEVEL GEARS. 

sin. B 



Tan. E — 



sin. B — 



No 



E' = E - B. 



USE OF BEVEL GEAR TABLE 

To use the bevel gear table, Table 21, divide the number of teeth in 
pinion by the number of teeth in gear; the quotient will equal the tangent of 
center angles. Find the nearest number in the table of tangents and on the 
same line on the left will be found the degrees, and at the top of the column 



BEVEL GEARS 137 

hundredth degrees for the center angle of pinion. On the same line of the 
right will be found the degrees, and at the bottom of the column hundredth 
degrees for gear. 

On the same line on the left will be found the angle increase, which, when 
divided by the number of teeth in the pinion, will give as a quotient the angle 
increase for either wheel. 

To obtain the face angles add the angle increase to the center angle, and to 
obtain the cutting angle subtract the angle decrement from the center angle. 
Now on the same Hne on the left will be found the diameter increase for the 
pinion, and on the same line on the right will be found the diameter increase 
for the gear. These when divided by the required diametral pitch equal the 
diametral increase for that pitch, which, added to the pitch diameters, give 
the outside diameters. 

AN ILLUSTRATIVE EXAMPLE 

In a pair of bevel gears, 24 and 27 teeth. 8 diametral pitch; 24 divided 
by 72 = 0.3333, which is the tangent of the center angle. The nearest 
tangent in the table is 0.3346, which gives: 

Center angle of pinion, 18.50 degrees. 
Center angle of gear, 71.50 degrees. 

On the same line at the left will be found the angle increase, 36, which divided 

36 
by the number of teeth in the pinion will give the angle increase — = i-5 

24 

degrees. This angle added to the center angle will give the face angle. 

The cutting angle is found by subtracting this angle, plus 16 per cent., 

from the center angle, which in this example would be 1.50 X 0.16 = 0.24, 

therefore the angle increment would be 1.50 + 0.24 = 1.74 degrees. 

Face angle of pinion = 18.50 + 1.50 = 20.00 degrees. 
Cutting angle of pinion = 18.50 — 1.74 = 16.76 degrees. 
Face angle of gear = 71.50 + 1.50 = 73.00 degrees. 

Cutting angle of gear = 71.50 — 1.74 = 69.76 degrees. 

On the same line to the left will be found the diameter increase for the pinion, 

1.90, which divided by the pitch, or ~—, = 0.237 inch. On the right of 

this same line will be found the diameter increase for the gear, 0.63, which 

divided by the pitch, or ~^, = 0.081 inch. 

o 

24 
The pitch diameter of the pinion is -^ = 3 inches. The pitch diameter 

72 
of the gear is -5- = 9 inches, 
o 

Outside diameter of gear = 9 + 0.081 = 9.081 inches. 
Outside diameter of pinion = 3 +0.237 = 3.237 inches. 



138 



AMERICAN MACHINIST GEAR BOOK 



For gears not at right angles, first determine the center angle, and enter the 
table for each gear separately, using the formulas given on pages 145 and 146. 
Except in the case of crown bevel gears, the formulas give tangents of the 
center angle, so the angles themselves may be most conveniently found from 
Table 21. 

Number of Teeth in Wheel 



13 

14 

IS 

16 
17 
18 
19 



23 
24 
25 
26 

27 
28 

29 
30 
31 
32 
33 
34 

35 
36 
37 
38 
39 
40 
41 
42 



42 



2" 38' 
3'' 2' 
2° 37' 
3^1' 
2° 35' 
^o 59' 
2° 35' 
2° 58' 
2° 34' 
2° 57' 
2° 33' 
2° 56' 

2° 31' 
2° 54' 
2° 30' 

2=52' 

2°2g' 

2° 50' 
2° 28' 

2° 49' 
2° 26' 
2° 48^ 
2" 2S'. 
2° 46^ 
2° 24' 

2° 44' 
2° 22' 

2%3' 
2° 20' 

2" 19' 

2° 39' 
2° 17' 

2° 37' 
2° 16' 
2° 36' 
2° 14' 

2° 34" 
2° 13' 

2^11' 

2° 10' 
2" 29' 

2" 27' 

2" 25' 
2° 6' 
2" 24' 

<s' 

"22' 
2° 20' 

2°l' 
2° 18' 

'o59' 

2° 17' 

i°S8' 
2° 15' 

1 57' 

2 13 



41 



2" 42 

3^6' 
2° 40 
3° 5' 
2° 38' 

3:4' 
2° 38' 
3° 2' 
2° 37 

2^35 
2° 59 
2 3Z 
2° 58' 
2° 32' 
2°S6' 
2° 30 
2° 54' 
2° 28' 
2° 52 
2° 27 

2° 26' 
2° 49 
2° 24' 
2° 47 
2° 23 
2° 46' 
2° 22 
2° 44 
2° 19 
2° 42 
2° 18' 
2° 40 
2° 17 
2° 38' 

2° 36' 
2" 13' 
2° 34' 
2° 12' 

2° 33' 

2° II' 
2° 31' 
2° 8' 
2° 29' 
2° 7' 
2° 27' 
2° 6' 
2° 26' 

2° 24' 
2° 2' 
2° 22' 
2°l' 
2° 20' 

I°59' 
2° 18' 

i°58' 
2° 16' 



40 



2" 45' 
3° 10' 
2° 44' 

2 43 

2 41' 
3° 6' 
2° 40' 
3° 4' 
2° 38' 
3^3' 

o37' 
3° I' 
2° 35' 
2^59' 

o33' 

°58' 

2° 32' 

2° 56' 

2° 28' 

:^< 

°26' 
2° 49' 
2° 24' 
2° 46' 
2° 22' 
2° 44' 
2° 21' 

2° 43' 
2° 19' 

'!4i'i 

^139' 
2° 16' 

2! 37' 

2 14' 

2o35' 

2° 12' 

2° 33' 
2° II' 

2° 31 

^y 

2 30 

2° 8' 

°28' 

°6' 

°26' 

:4' 

2 24 

^:3' 
22 

2°l' 
2° 20' 



39 



3" 15 
12^47 
3" 13 

2° 46 

|3°12 
12° 44 
|3°io 
2° 43 
3° 8' 
2° 42 

3:7' 
2 40 

3° 5' 
2° 38 

3:3' 
2° 37 
3° I' 
2° 35 
2° 59 
2° 33 
2^58 
2° 32 
2° 56 
2° 30 
2° 54 
2° 28 

^152 
2° 26 
2° 49 
2° 24 
2° 47 
2 23 

2° 45 
2° 21 
2° 44 
2° 19 
2° 42 
2° 18 

2° 39 
2° 16 

2° 37 
2° 14 
2° 36 
2" 12 

2° 34 
2° II 

2° 32 

^y 

2 30 

2° 8' 
2° 28 
2" 6' 
2" 25 

<5' 
2° 23 



38 



53 
3" 19 

52 
3" 18 
2" 50 
3° 16 

2%8 

3° 14 
47 

3" 13 
45 

3" II 

2° 43 

3>' 

2 42 

3:7' 

2° 40 

3:5' 
2° 38 

3:3' 

2° 37 
3°l' 
2° 35 
2° 59 
2° 33 
2° 58 
2° 31 
2° 56 
2° 29 

<S4 
2 27 

2° 26 

2° 49 
2 24 
2° 47 
2 22 
2° 44 
2 20 
2° 42 
2° 18 
2° 40 
2° 16 
2" 38 
2° 14 
2° 36 
2° 13 
2° 34 
2° II 
2° 32 

^°9' 
2° 30 
2° 7' 
2° 2 



37 



2 57 
3° 24 
2° 56 
3° 23 
2° 54 

3 21 
2° S3 
3 19 

3° 17 
2 49 

3^5 
2 47 

2 45 
3° 12 
2° 43 
3° 9' 

42 

3-7' 

40 

3; 5' 
2° 38 
3° 2' 
2° 36 
3" I' 
2° 34 
2° 58 
2" 32 
2° 56 
2° 30 

2° S3 
2° 28 

2° 26 
2»49 
2° 24 
2° 46 
2° 22 
2° 44 
2° 20 

2° 43 
2° 18 

2° 40 
2° 17 

2° 37 

2 35 
2° 13 
2*34 
2° 12 
2° 32 



36 



3"2' 
3° 29' 
3°o' 
3° 28' 
2° 58' 
3° 26' 
2" 56' 
3° 24' 
2° 55' 
3 22' 
2°53' 
3 20 

2° 51 

3 17 
49 
3" 16' 

:47 

3 14 

:45 

3 II 
2° 43 
3° 8' 

142 
3° 6' 

:3? 

3 4 
37 
3-1' 
2° 35 
2 59 
2 33 
2 57 
31 

';s4 

52 

"27 

2 49 
2 24 
2° 47 
2 23 

145 
21 

I 42 
2° 18 
2° 40 
2° 17' 
2° 38' 
2° IS' 
2° 36' 



35 



3° 6' 
3° 35 

3o4' 
3 33 
3° 3' 
3 31 
3° I' 
3 29 
2° 58' 
3° 26' 
2° 57 
3 24 
2^55' 
3 22 

2° 53' 

3o'9" 
2° so' 

2° 48 

3:^5 
2 46' 
3° 12 
2° 44 
3° 10 
2° 42 

3^7' 
2° 40 

3^4' 
2° 38' 
3° 2' 
2° 36' 
3°o' 
2° 33' 
2° 58' 

2° 31' 
2° 54' 
2° 29' 

2° 52' 
2" 27' 
2° 50' 
2° 25' 
2° 48' 
2° 23' 
2° 45' 
2 21 
2" 42 
2° 19 
2° 40' 



34 



3" II' 
3° 40' 

^y 

3° 38' 

3° 7' 
3° 36' 

3:4', 
3 34 

3° 31' 
3° I' 

3° 27' 
2° 58' 
3° 26' 

'>: 

3„ 24 
2° 54' 

3^21' 

2° 52' 

3:^9' 
2° 49' 

3 16' 
2° 47' 

3^4; 
2^45 
3° II' 
2° 43' 
3° 8' 
2° 40' 
3° 6' 
2° 38' 

2° 36' 
3°o' 
2° 34' 

'o57; 

2° 32' 

2° 55' 
2° 29 

2° 53 
2° 27 

2° 25 
2° 48 

2°23' 

2° 45' 



33 



3 15' 3' 
3" 46' 3' 
3° 14' 3' 



32 



31 



3 44 
3° 12' 
3" 42' 
3° 10' 
3° 39' 
3 7', 
3 37 
3° 5' 
3° 34' 
3° 3' 
3° 31' 
3°o' 
3° 29' 
2° 58' 
3° 26' 
2° 56' 
3° 23' 
2*53' 
3 20' 
2° 51' 
3° 18' 
2" 48' 

3° 15' 
2° 46' 
3° 12' 
2° 43' 
3^9' 

2''4l' 

3° 6' 
2° 39' 

2° 37' 
3° I' 
2° 34' 
2° 58' 
2° 32 
2° 56' 
2° 29 

2° S2, 
2° 27 
2° 51 



21 

52' 

19' 

50' 

17' 

47' 

14' 

45' 

12' 

42' 

9' 

39' 

7' 

36' 

4' 

33' 

2' 

31' 
o' 

27' 

57' 

24' 

54' 

22' 

52' 

19' 

49' 

16' 

47' 

13' 

44' 

10 

42 

7' 

39 

5' 

36' 

i' 

34 

S8' 

32 

S6' 



30 



3 33 
4° 6' 

24'i3° 29' 
4° 3' 
3° 27' 
4°o' 
3° 24' 
3 57' 
3" 22' 

48' 3° 5^ 



56' 

22 

54 

19 

51 

17 



3° 50 
3° 16' 
3" 47' 
3° 13' 
3° 44' 
3° 10' 
35' 3° 40' 
3' 3" "' 



14 

45' 

12' 

42' 

9' 

38' 

6' 



32' 

o' 

29' 

58' 
26' 
55' 



3-36' 
3° 4' 
3° 32' 
3° 2' 
3° 30' 
2° 58' 



23 3 27 
'S3' 2° 56' 
' 2o'|3 24 
49'j2°53 
16' 3 20' 
47' 2° so- 
I3'l3 17 
44', 2° 47 

^°',K '4; 

42 ,2 44 
7' 3° 10' 



29 



2 42 
3° 7' 



3° 38' 
4° 13' 
3° 36' 
4° 10' 
3° 33' 
4° 7' 
3° 30' 

■° a' 

%4 , 
3 27' 
4°o' 
3° 24' 

3:56; 
3 21' 
3° S?.' 
3° 18' 
3° 49' 
3^5; 
3 45' 
3° 12' 
3° 41' 
3° 8' 
3° 38' 
315' 
3° 34' 
3° 2' 

3° 31' 

3° 28' 
2° 56' 

3„ 24 1 
2° S3' 
3° 20' 
2° 51' 
3° 17' 
2° 47 
3° 14 



28 



3" 45' 
4° 21' 

3° 43' 
4° 17' 
3° 39 
4° 13' 
3° 36' 
4 10 

3° 33' 
4° 6' 
3" 29' 
4° 2' 
3° 26' 
3° 57' 
3 23' 
3° 55' 
3° 19' 
3° 51' 
3° 17' 
3° 47' 
3° 13' 
3° 43' 
3° 9' 
3° 39' 
3° 6' 

3° 35' 

3° 32' 
3°o' 
3° 28' 
2° 57' 
3° 24' 
2° S3' 
3°2i' 



Table 20 — Addendum and Dedendum Angles for Bevel Gears. Angle Between 



BEVEL GEARS 



139 



Number of Teeth in Wheel 



27 



3" S3' 
4° 29' 
3° 49' 
4° 25' 
3° 46' 
4 21' 
3° 43' 
4° 16' 

3° 39' 
4° 13' 
3° 35' 
4° 9' 
3° 32' 
4^4' 
3° 28' 
4° I' 
3° 24' 
3° 56' 
3° 21' 

3° 17' 
3° 48' 
3° 14' 
3° 44' 
3° 10' 
3° 39' 

^y 
3° 36' 

3° 32' 

3°o' 
3° 28' 



26 



4 o 

4° 37 
3° 57 
4° 33, 
3° 54 
4 29 
3° 49 
4 24 
3° 45 
4 

4° 16 

3° 37 
4 10 

3° 33 

3 29 

4^2' 
3° 25 
3° 58 
3° 

3° 52 
3° 

3° 49 
^0^4 
3° 44 
3° 10 
3° 40 
3° 7' 
3° 36 



4:7 
'4° 46 

;4:3' 

4 42 
'3° 59 
'4° 37 
'3° 55 
' 4 32 

:<S2 
4 27 

>:47 
4 23 
' 3° 43 
' 4° 18 
' 3° 38 

3 34 

4° 8' 
' 3° 30 

'3° 26 
' 3° 58 
'3° 22 
'3° 54 
"3° 18 
''3° 49 
'3° 14 
'3° 45 



24 



4° 16' 

4° 56' 
4° 12' 

4° si' 

4:7' 
4° 46' 

4° 2' 
4° 41' 
3°S8' 
4° 36' 
3° 53' 
4 30' 
3° 49' 
4 25' 
3° 44' 
4 19' 
3° 40' 
4° 14' 
3° 35' 
4° 8' 

3° 31' 

4^4' 
3 27' 
3° 59' 
3° 22' 
3° 54' 



23 



25' 

6' 



20' 1 4' 



o' 

15' 

54' 

9' 

49' 

5' 

44' 

o' 

37' 
55' 
32' 
50' 
26' 

45' 

20' 

40' 

14' 

36' 

9' 

32 

4' 



34' 
16' 

29' 
10' 

24' 

4' 

18' 

58' 

14' 

52' 

8' 

45' 
2' 

39' 

56' 
33' 
51' 

46 3 

2l'4' 

41'' 
15' 



43 
28' 

38' 
21' 
32' 
14' 
26' 

7'- 
20' 
o' 
13' 

54' 
8' 

47' 

2' 

40' 

56' 

34' 

52' 

27' 



4" 55 
5° 40 
4° 48 
5 32 
4° 40 
5° 25 
4° 34 

4° 28' 
5° 10 
4° 22 

5^' 
4 15 

4° 55 
4° 8' 
4° 48 

4° 3' 
4° 41 



19 



18 



5° 16 
6° 6' 
5° 8' 

5° 57 
5°o' 
5° 48 
4° 52 
5° 38 
4 45 
5° 29 
4° 37 
5° 21' 
4° 29' 
5° 12' 



17 



16 



15 



S" 42' 5" 56' 
6° 36' 6° 52' 
5" 32' 5° 45' 
6" 24', 6° 39' 



14 



13 



5° 22' 
6° II 
5° 12 
6° 2' 
5° 3' 
5° 50 



6° 10' 6° 27' 6° 43' 
7° 9' 7° 27' 7° 46' 
5°57':6°i2 
6° 54' 7° 10 



5 37'iS"4S 
6° 24' 6° 43' 
5° 2 
6° 14' 



J = Angle LuGODeltiEkesat 




12 
13 
14 
15 

16 

17 
18 

19 
20 
21 

22 

23 
24 

25 

26 
27 
28 
29 
30 
31 
32 

33 

34 
35 
36 
37 
38 
39 
40 

41 

42 



^ 

S 



^ 



Axes 90°, Tooth Proportions Brown & Sharpe Standard. By F. Withers 



140 



AMERICAN MACHINIST GEAR BOOK 



Number of Teeth in Wheel 



13 



14 



IS 

16 



17 
18 



19 



23 



24 



25 

26 

27 
28 



20 



30 



31 



32 



33 



34 



35 
36 
37 
38 



39 



40 



41 



42 



J. 
K. 

J. 

K. 

J. 
K. 

J. 
K. 

J. 
K. 

J. 
K. 

J. 
K. 

J-. 
K. 

J. 
K. 

J. 
K. 

J.. 
K. 
J.. 
K. 

J.. 

K. 

J.. 
K. 

J. 

K. 

J.. 

K. 

J.. 
K. 

J. 

K. 

J. 

K. 

J.. 
K. 

J. 
K. 

J.. 
K. 

J. 

K. 

J. 
K. 

J. 
K. 

J. 
K. 

J. 
K. 

l. 

J. 

K, 

J. 
K. 

J. 
K 



72 



"34 

:49 

34 
°49 
°34 
°48 

°48 
33 

°47 

o47 

o47 

33 

I 47 

32 

°46 

32 

°46 

o4^ 

31 

>5 

c3° 

o44 

o3° 

44 

o3° 

44 

c43 

0^9 

42 

° 28 

°42 

-28 

:4- 

o4i 
0=7 
o40 

°26 

°40 

°26 

o39 

38 
0^4 
38 
0^4 
38 
23 
37 
23 
36 
23 
35 



71 



70 



I" 3-;' 


I" 37' 


I" 38 


1° 50' 


I°52' 


i"54 


1° 35' 


I" 37' 


I" 38 


i"-;o' 


1° 52' 


i" 53 


l" 35' 


^l37' 


I" 38 


I 50' 


I" 51' 


i" 53 


^0 35' 


i"36' 


^u3« 


i"so' 


i°5i' 


^0 •■'3 


I" 3^' 


i" 36' 


I" 37 


i"4g' 


^^i; 


l"52 


I 34' 


i\^^' 


I 37 


i"40' 


1° 50' 


I" 51 


^:34; 


I 35' 


I 37 


l"40' 


I 50' 


I 51 


^!34' 


^:3.^' 


I 36 


1-48' 


I 50' 


^0^^ 


I"3S' 


I 34' 


I" 36 


I" 48' 


I 49' 


i"5i 


i°33' 


^::34; 


^35 


'o47; 


1 49' 
1 34' 


I 35 


1° 47' 


i°48' 


I 49 


i" 33' 


I^^v 


"■l^'- 


^:46; 


I" 48' 


I" 49 


I" 32' 


i°33' 


I 34 


I" 46' 


i''48' 


i„ 49 


l"S2' 


I" 3V 


I 34 


I" 45' 


K< 


i"48 


I 31' 


I" 32' 


i" 33 


^>s: 


i°46' 


^47 


i"3i' 


I°32' 


I 33 


^u4S' 


1° 46' 


^"47 


I 30' 


I" 31' 


i" 33 


^"44' 


^:45' 


i"46 


I" 30' 


1° 31' 


l"32 


^:< 


^"45' 


I 46 


i" 20' 


i" 30' 


I 32 


^:42; 


1-44' 


<4S 


^ ^9 


I 30' 


^o3i 


1 42 


i„44' 


^o4S 


l"28' 


^"3°! 


<30 


1° 42' 


I 4V 


i>4 


l"28' 


I" 29' 


1^30 


'>^; 


l°43' 


i„44 


i" 27' 


I 29 


I 30 








1 41' 


I 42' 


^0 43 


1 27' 


l"28' 


I 29 








1 40' 


I 42' 


^o43 


t"26' 


l"28' 


l" 28 






,0 


1 40 


I 41' 


1 42 


l"26' 


I 27' 


l"28 








i 39 


1 ^0 40' 


\ 4i 


I 25' 


l''26' 


i" 27 








i 39 


1 ^ 40, 


i„ 41 


I" 2^' 


Il"2=;' 


I- 27 


1° 38' 


li°39' 


I 40 


1° 24' 


ii° 25' 


I" 26 


i"37 


^: 38; 


^"39 


l"24 


l"2^' 


i" 26 


1° 37 


I" 38 


i"39 


1° 24 


1° 24 


^u^.S 


I" 36 


,^"37 


I 38 



69 



63 



I 40 

I 40 

I 39 

I 39 

^38 
^! 54 
^38 
^53 
^38 
^53 
i„37 
^53 
^37 

^37 
i°52 
I 36 

i°36 
'I 51 
^35 
^5o 
^35 

I 34 
I 49 

^34 
I 49 

I 33 
i°48 

i°33 
^47 
I 33 
^47 
I 32 
1° 46 
1° 32 

'o45 
^31 
I 45 
1° 30 

i°45 
I 30 
i°44 
I 30 

^:43 

I 29 
i°43 
1° 28 
1° 42 

I°28 

^:4- 
1 27 

1° 40 

1° 26 

1° 40 

I°26 

i°39 



67 



I 41 
1° 57 
^141 

^^41 

I 40 

I 40 

^°55 
I 40 

1 39 

I 39 
^o54 
^38 
^^54 
^38 
^l53 
^38 
1,52 
^37 
I 52 
^37 

i°36 

^35 

^35 
^5° 
^o35 
i°49 

'o34 
i°48 

^o34 
i''48 

i°33 
^47 
I 33 
^47 
I 32 
i°46 
1° 32 
i°46 

^:3^ 

^45 
I 30 
i°44 
I 30 
l°44 
I 30 

^:43 

I 29 
l''42 

I°28 

^^41 

,1 27 

h:4^' 

]l 27' 

'1° 40 



66 



'«43 
i°S8 
i°43 
i°58 
^=43 
i°58 
1° 42 

^:57 

I 42 

^57 

^57 
^!4i 

I 40 
I 40 

I 40 

'o55 
i°39 
^54 
I 39 
'I 54 
I 38 
^=53 
^38 
I 52 
1^37 
^52 
I 37 

i°36 
^«5i 
^35 

^35 

^35 
I 49 

^l34 
i°48 

i°33 
i°48 

i''33 
^47 
I 32 
i°46 
1° 32 
i''46 

^:3^ 

i°45 
I 30 

^:4s 
1 30 

i°44 
I 30 

^:43 

I 29 
I" 42 

1° 28 

i°4i 



6S 



I" 45 
2° o' 

i°44 
2°o' 
i°44 

2°o' 

i°44 
^59 
^43 
l°58 

'o43 
i°S8 

^o42 
i°58 
i" 42 

^:57 

I 41 

^57 
I 41 

I 40 

I 40 

1 39 

^54 
I 39 
^l54 
^38 
^o53 
^38 
^53 
I 37 

'o52 
i°36 

^35 
I 51 
I 35 

^34 

^34 
I 49 
I 33 
i°48 

l°33 

^:47 

I 32 

^:47 

1-3 

I 

I 
I 

I 
I 
I 
I 
I 
I 



64 



'46 
'31 
'46 
'31 

'45 



42 



i°46 

2° 2' 
i''46 

2 2 

i°43 

2°l' 

i°45 
2 o 

2''o' 

i°45 

2°o' 

i°44 
^59 
^43 
^59 
I 42 

^:59 

I 42 
i°58 
1° 42 

^:57 

^41 
^57 
^41 

I 40 
I 40 

1 39 

^o55 
i°38 
^oS4 
^38 
^53 
I 37 
1° 53 
I 37 

^36 

^35 
i^So 
i°35 
^5° 
I 35 
I 49 

^o34 
i°48 

i°33 

^:47 

I 32 

^147 
I 32 

i''46 

1° 32 

i°45 

^31 
I 44 
i°30 
i°43 



63 



62 



1° 48' 1° so' 
2° 4' 2" 6' 
i°48'i°So' 

2° 4' 2°S' 

i°47'|i so' 
2° 3' |2°5' 
i°47'|i°49' 
2° 3' 2° 5' 
i^e' i°48' 
2° 2' |2°4' 
1° 45' i°48' 
2° 2' 2° 4' 
i°4S'l°47' 
2''i' 2" 3' 
I°45'k47' 

2° l' |2°2' 

i°4S'|i°46' 
2-0' 2° 2' 
i°44'|i°45' 
2°o' 2°l' 

^: 44', 1:45' 

I 59,2 I 
i°43'i°45' 
58'!2°o' 
42' 1° 44' 



6i 



58' 

:42: 
1° 57 

i°S6' 
i°4o' 
i°56' 
i°4o' 
i°5S' 



2"0' 

^143; 

^59, 
I 43 
i°58' 

I°42' 

i°S8' 
i°4i' 
i°57' 



^.39 |i 40 

i°S4'i°56' 

i°38'i°4o' 

i°S4'-° - 

i°38' 

^o53' 

i°38' 

i!52' 



60 



59 



I 37 
i°5i' 
i°36' 

^35, 
'I 50 
^35 
I 49 

'I 34' 

i°48' 

i°33' 

i°48' 

1° 33' 

i%7' 

i°33' 

i°46' 

1° 32' 

i°45' 

1° 31' I" 32' 

I 44',! 46' 



55 
i" 40 

i°55 
1° 39 
i°54 
i°38' 

K 53, 
i°38' 
i°52- 
^37 
i°S2 
i°36' 
1° 51' 
I 35' 
i°5o' 
i°35' 
i°49' 

i°34' 
i°48' 

1° 33' 
i°48' 

i°33' 
i°47' 



i^Si' 
2° 8' 

i°Si' 
2° 7' 
i°5o' 
2° 7' 
i°5o' 

i°5o' 
2° 6' 
i°5o' 
2° 6' 
1° 49' 
2° 5' 
i°48' 
2W 
i°48' 

2° 4' 
i°47' 
2° 3' 
i°47' 
2° 2' 
i°46' 
2° 2' 
i°45 

2°l' 

i°45' 

2°o' 

i°44' 

i°59' 

i°43' 

i°59' 

i°43' 

i°58' 

I°42' 

I'^S?' 
i°4i' 
i°56' 
i°4o' 
i°56' 
1° 40' 

1° 55' 
1° 40' 

i°54' 
i°39' 
i°53' 
i°38' 
i°53' 

^:37; 

I 52 
i°36' 
1° 51' 
i°35 
i°5o' 
i°35' 
1° 49' 

^:34 

I 49 

i°34'ii"35' 
l°48' i°49' 

1° 33';!° 34' 
l°47'ii°48' 



I-S3' 

2° 10' 

i°53' 
<9', 

2° 9' I 

1° Si'i 

2° 9' 

i°5i' 
2° 8' 
1° 50' 
2° 8' 

2° 7' I 

2° 6' I 

1° 49'i 

2° 6' 1 

i°49' 
2° 5' 
i°48' 
2° 4' 
i°48' 
2° 4' 
i°47' 
2° 3' 
1° 46' 
2° 2' 
i°45' 

2°l' 

i°45' 
2° i' 

i°44' 

2°o' 

i°44' 

1° 59' 

i°43' 

i°58' 

1° 42' 

i°58' 

i°4i' 

i°57' 

i°4o' 

i°56' 

1° 40' 

1° 55' 

i°39' 

i°55' 

i°38' 

i°54' 

i°38' 

i°S3' 

1° 37' 

I°S2' 

i°36' 
i°5i' 
i°35' 
i°5o' 



i°55' 
2° 12' 

i°55' 

2° 12* 

i°54' 
2° 11' 

i°54' 
2° 11' 

|i°53' 

2° 10' 

i°53' 
2° 10' 

i''52' 

|2°9' 

ll° 52' 

2° 8' 

i°5i' 
2° 8' 

,1° 50' 
2° 7' 
1° 50' 
2° 6' 
1° 50' 
2° 5' 

,i°49' 
2° 5' 
i°48' 

2° 4' 
i°47' 
2° 3' 
i°46' 
2° 2' 

1=45' 
2° i' 

1° 45-' 

2°l' 

i°45' 

2°0' 

i°44' 
i°59' 
i°43' 
i°58' 

I°42' 

i°57' 
i°4i' 
i°56' 
i°4o' 
i°56' 
1° 40' 
i°55' 
i°39' 
i°54' 
i°38' 
i°53' 
i''37' 
i°52' 
i°37' 
i°Si' 
i°36' 
i°5o' 
i°35' 
i°49' 



Table 20— -Continued, Addendum and Dedendxjm Angles tor Bevel Gears 



BEVEL GEARS 



141 



Number of Teeth in Wheel 



58 



2" 14' 



57 



i°S8' 
2° 16' 

I°S8 



S6 



55 



2* 14' 2° 16' 
i°5S'!i°58' 
2° 13' 2° IS 
1° SS' i" 57 



2" IS' 
i°56' 
2° 14' 
i°56' 
2 14' 

13' 
I" 55' 
2" 12' 

l°S4' 



2" 13 

I' 55 
212 

2 12 

^°54 
2° II 

2° 10 

2° 10' 

1° 52' I- 54 
29' 2° II 

i°Si'i°53 
2" 8' 2° 10 

i°5o'i°53 

2 7' 2° 9' 

'I SO- " 

2° 7' 

l%9 

2 

I 
2 



2°0' 
2° 19 
2°o' 
2° 18' 
2°o' 
2° 18' 

2 17 

i°S8' 
2° 17' 

i°57' 
16' 
i°S6' 



2° 8 



»6' 

:^'' 

i" 48' 
2° 4' 

i*'47' 
2° 3' 
i°46' 
2° 3' 
l°4S' 

2° 2' 

l°44' 

2° l' 

l°44' 

2''o' 

I°43' 
^59; 

l''42' 

l»4i' 

i°S7' 

1° 40' 

I''S6' 

1° 40 

i°56' 

l''39 

i°S5' 

i°38' 

i°S3' 

i°38' 

i''S2' 

i°36' 

i°5i' 

i°36' 

i°So 



'50' 

'50' 

"6' 

°49' 
5' 

>8' 
2W 
i°47' 

3' 
i°46' 
2° 2' 

°45' 
2° 2' 
i°4S' 

2°o' 

i''44' 

^ 59; 

I 43 

i''58' 

i°42' 

i°57' 

i°4i' 

i°57' 

1° 40' 

i»56' 

1° 39' 

1° 39' 

i°53' 
i°38' 
i°52' 
i°37' 
I'Si' 



15 
56' 
14' 

2" 14' 

2° 13' 
2° 12' 
2°H' 

^:s3: 
2" 10' 

2 10 

i°5i' 

2° 8' 
i°5o' 

2° 6' 
i°49' 

i°43' 

'o4' 
i°46 
2° 3' 
i°46' 
2° 2' 

2° i' 

2° o 

i"'44' 

^059; 
I 43 
i»58' 
^o42 

KS7 
hi 41' 
i°56' 

^:4°; 
1^ 53, 
1 39 

i°54' 
i°38' 
i°S3 



54 



2 21 
2° 2' 
2° 21' 

2°l' 
2° 20' 
2°o' 
2° 19' 
2°o' 
2° 19' 
2''o' 
2° 18' 
1^59' 

I 59 
16' 
58' 

2" 16' 

57' 

is' 

1-56' 

"14' 

"56' 

2° 13' 

"SS' 

°I2' 

!S4' 
12' 

°S3' 
2° 11' 

10 

°S2', 
2»9' 

i°5i' 

:-%■ 

2»6' 

i°49' 
2° 5' 
i°4S' 
2' 4' 
i°47' 
23' 
i°46' 
2° 2' 
i°45' 

2=1' 

i°44' 

2°o' 

1° 43' 

I'SS' 
1° 42' 
i-'S?' 
i°4i' 

1 40 

I'SS' 

1° 39' 

i°54' 



53 



4' 

12° 24' 

12° 4' 

2° 23' 

2° 3' 
2" 22' 

2° 3' 
2° 21' 
2° 2' 
2° 21' 
2° I' 
2° 20' 
2''o' 
2° 20' 
2°o' 
2° 19' 
2°o' 
2° 18' 

2 17' 

i°58' 
2° 16' 
i°57' 

I"S6' 

'1^4; 

2° 14' 
i°S5' 
2° 13' 
'I 54; 
2° 12' 

-:s3: 
2° 11' 

^oS2' 
2° 10' 
I°Sl' 
2° 9' 

i°Si' 

2° 8' 

1° so", 
27' 
l°49' 
2° 6' 
i°48' 

i°47' 

i°46 
2° 3' 
i°45' 

12° 2' 

ho 44' 
2°0' 

ho 43' 

ho 4^', 

I 41' 
i°57' 
i°4o' 

y S6' 



52 



2° 6' 
2° 26' 

"o5' 
2" 26' 

2° 25' 

<s' 

2^24' 

2 24 
2° 3' 
2° 23' 
2° 3' 
2° 22' 
2° 2' i 
2° 21' 

2°l' ! 
2° 20' 
2 O 
2° 19' 
2°o' 
2° 18' 

^:59: 

i°5S' 
2° 16' 
1-57' 
2° 16' 

'o5S 

2° 14' 

2° 13' 

'o54; 
2° 11' 

^:s3: 
2° 10' 

I°52' 

':«', 

i°5i' 
2° 8' 
1° 50' 

2° 6' 
^l 49' 

1 48 

2 4 I 
I 47'| 
2° 3' 
1° 46' 

2°l' 

i''45' 

2°0' 

1^44' 

i°59' 
i°43' 
i°S8' 
1° 42' 
l''S7' 



2° 8' 
2° 29' 
2° 8' 
2° 28' 

2° 7' 
2° 28' 
2° 6' 
2° 27' 

2° 5' 
2° 26' 

2° 2S' 
2»5' 
2° 24' 

'I 4' 
2" 23' 

K^' 

2' 23' 

2" 2' 

2" 22' 



2- 21' 
2°o' 
2" 20' 
2°o' 
2° 19' 

'IS9' 

2" 18' 

i"58' 
'1^7' 
'I 57 
2° 16' 

^:s6; 

2 14' 
i^SS' 

2° 13' 

^: 54; 
2° 11' 

2° 10 

I°52' 
2° 9' 

i°5i' 

2° 8' 

1° 50' 

2° 7' 

i°49' 
2° 6' 
i°48' 
2° 5' 
i''47 

2^3' 

i°46' 
2° 2' 
i''45 
2°i' 
ll''44 

'2°o' 

i°43 
ii°59 



51 



50 



49 



48 



2" II' 
2° 32' 
2° 10' 
2° 31' 
2° 10' 
2" 30' 

2 29 
2° 8' 
2° 28' 
2° 8' 
2^27' 
2" 7' 
2° 26' 
2° 6' 
2° 25' 

2° 24' 

'I 4' 
2»24' 

2" 23' 

2° 3' 
2° 22' 
2° 2' 
2»2l' 
2° I' 
2" 20' 

2" o' 

2" 19' 

'I 59' 
2" 18' 

i''58' 

'I 57' 
2° is' 
i°56' 
2° 14' 

fo 55;, 
2 13 I 

2° 12', 

1:53', 
2 ii'i 
i" 52', 
2° 9' I 

I Si'i 
2° 8' 
1° 50' 
2° 7' 
i°49' 
2° 6' 
i°48' 
2° 4' 
i°47' 
2° 3' 
i°46' 
2° 2' 
"45' 
i' 
44' 
o' 



15 
'"I 35; 
2 14 

2° 34' 
2° 13' 
2° 33' 
2° 12' 
2° 32' 
2° 10' 

2° 31' 
2° 10' 
2° 30' 
2° 9' 
2° 29' 
2° 8' 
2° 28' 

2»7' 

2" 27' 
2" 6' 

2" 26' 

'I 5' J 
2° 25' 
2:4' 

2 24' 

2° 23', 
2° 2' 
2° 22', 

2°l' 
2° 21' 
2''o' 
2" 20', 
2°o' I 
2° 19' 

^l 59'i 

l^ 58' 
2° 16' 

i:s7' 
2° 15' 

i°S5', 
'1^4' 

^^55; 

'x°5!'' 

h:53 

2 10 

h°s2-' 

h:9' 
,1 50 

2° 8' 



I 49 
2° 6' 
i°48' 

2=5' 

I°47' 
2° 4' 
i°46' 
2° 3' 
i°4S' 

2°!' 



2° 


16' 


2° 


38' 


2° 


15' 


2'^ 


37' 


2" 


15' 


2" 


36' 


2" 


14' 


2" 


35' 


2" 


13' 


2" 


34' 


2" 


13 


2*^ 


33' 


2" 


12' 


2" 


32' 


2" 


11' 


2" 


31' 


2" 


10 








30 






2 


<4 




29' 


2" 


8' 


2° 


28' 


2° 


7' 


2" 


26' 


2° 


6' 


2° 


25' 


2" 


< 


2" 


24' 


2" 


4' 


2" 


23' 


2" 


3' 


2" 


22' 


2° 


2' 


2" 


2t' 


2° 


l' 


2" 


19' 


2" 


0' 


2° 


18' 


l" 


59' 


2'' 


17' 


l" 


58' 


2" 


16' 


l" 


57; 


2" 


IS' 


l" 


55' 


2" 


13' 


l" 


54' 


2" 


it' 


1° 


52' 


2" 


to' 






1 


Si 


2" 


9' 


l" 


so' 


2° 


7' 


l" 


49' 


2" 


6' 


1° 


48' 


2" 


5' 


l" 


47' 


2" 


4' 


I^ 


46' 



47 



2" 19' 2 
2° 41' 2 
2° 18' 2 
2" 40'i 2 
2° I7'j2 
2° 39'i2 
2° 16', 2 
2° 38', 2 
2° IS'|2 
2 37 2 
2° 15' 2 
2° 36', 2 
2" 14' 2 
2° 3S',2 
2 13', 2 
2° 34', 2 
2° 12' 2 
2° 33' 2 
2° 11' 2 
2° 31' 2 
2° 10' 2 
2° 30' 2 
2° 9' 2 
2° 29', 2 
2° 8' 2 
2° 28' 2 
2° 7' I2 
2° 27 
2° 6' 
2° 26'] 2 
2° S' 2 
2° 24' 2 
2° 3' 2 
2° 23' 2 
2° 3' 2 
2° 22' 2 
2° 2' I2 
2" 20' 2 
2°0' 
2°i9' 

i°59' 
2" 18' 
i°58' 
2° 17' 
i°S7' 
2° is' 

^:56: 

2 13 

1° 54' 
2 12' 

i°S3' 
2° 11' 2' 

i''52'i' 
2° 9' 2' 
i' 51' i' 

2° 8' 2' 

i°5o'i' 
,0 „/ .t 

27 2 

i°49'i' 
2° 5' '2< 
i°47'i' 
2° 4' 2' 



21 2" 

44' 



46 



21 

43 
20 
42 
19 
41 
18 
40' 2 



45 



24' 2" 27 
47' 2° 51 
24' 2° 27 
46' 2° 50 

23'|2'' 26 

45'|2°49 
22' 2 2S 
44' 2° 48 

22'[2° 24 
43' 2° 46 



44 



17' 
39' 
16' 
38' 

15'|2 

37;|2 

14 

36' 

13' 

35' 

12' 

35' 

11' 

31' 

10' 

30' 

9' 

29' 



23' 

7' 
26' 

5' 

25'|2- 
4 |2 
24'i2' 

3' h' 

22', 2' 

2' I2' 

2l'|2' 
l' 2' 
2o'i2' 

o' 2' 

is';2' 
59'' 2' 

I7',2' 

15^2 
57,1 

55,1^ 

13 |2 

S4',i' 
II' 2' 

S2',l' 

si' I 
8' 2' 
' so' i" 
'6' 2' 
'49'!' 
' 5' 2- 



21' 
42' 
19' 
41' 
18' 
40' 
17' 
39' 
15' 
37' 
15' 

35' 
13' 
34' 
12' 

33' 
11' 
32' 

10' 2° 12 
30' 2° 32 
9' 2° II 
29 2 31 
8' 2° 10 
28' 2° 30 
6' l2°8' 
26' 2° 28 

5' '2»7' 
25' 2 27 

4' |2°5' 
23' 2° 25 

3' 12° 4' 
22' 2° 24 



2 23 

2° 45 
2° 21 

2° 44 

2° 20 

2° 43 
2° 19 
2° 41 

2°lS 

2° 39 
2° 17 
2^38 
2° 15 
2° 37 
2° IS 
2° 36 
2° 13 
2° 34 



I 

20' 

o' 

19' 

59' 

17' 

S8' 

16' 

57' 

15' 

55' 

13' 

S2,' 

11' 

52' 

10' 

51' 
8' 
50' 
7' 



^3- 
2 22 
2° 2' 
2° 21 

2°l' 
2° 19 

^!59 

2° 18 

i°58 
2^16 

^:57 

2 14 

^:55 

2 12 

^IS4 

2 II 

i''53 
2° 10 

I°52 
2° 8' 



43 



2-30' 

':s4: 
2 30 

2" 53' 
2° 29' 

2^52' 
2° 28' 

2 27 

2° so' 
2° 25' 
2° 49' 

2»25' 

2»47' 

2° 23' 
2" 46' 
2° 22' 

2° 44' 
2° 21' 

2° 43' 
2° 19' 
2° 42' 
2° 18' 
2° 40' 
2° 17' 
2° 39' 
2° 15' 
2° 37' 
2° 14' 
2° 35' 
2° 13' 
2° 34' 
2 12' 
2° 32' 
2° 10' 
2° 31' 
2° 9' 
2° 30' 
2° 8' 
2° 28' 
2° 6' 
2° 26' 

2° 5' 
2 25' 
2° 4' 
2° 23' 
2° 3' 
2° 21' 
2°l' 
2° 20' 
2°0' 
2° 18' 

i°58' 
2° 16' 

i°57' 
2° 14' 

i°55' 
2° 13' 
i°S4' 
2° 11' 

1° 53' 

2° 10' 



2 34 
2° 58' 
2° 33' 
2° 57' 
2° 32' 
2° 56' 
2° 31' 
2^54' 
2° 29' 

2° 53' 
2° 28' 
2° 52 
2° 27 
2° so 
2° 26' 

2° 49 
2° 25 

2° 47 
2° 24 
2° 46; 
2° 22' 
2° 45 

2° 21 

2° 43 

2° 19 
2° 41 
2° l8 
2° 40 
2° 16' 
2^38' 
2° IS' 
2° 36' 
2° 14'' 

2° 34' 
2° 12' 

2° 33' 
2''H' 
2° 32' 
2° 9' 
2 30' 
2° 8' 
2° 28' 

'17' 

2 27 

2° 5' 
2° 25 

2° 4' 
2 23 

2° 3' 
2° 22 
2° I' 
2° 20 

^^59 
2° 18 
i°58' 
2° 16' 
^IS7 

^56' 
2 13 

^:5s' 
2 II 



13 
14 

IS 
16 
17 
18 

19 
20 
21 
22 
23 
24 
25 
26 

27 
28 

29 
30 
31 
32 
33 
34 
35 
3^ 
37 
38 
39 
40 
41 
42 



^3 



Angle Between Axes 90°, Tooxn Proportions Brown & Sharpe Standard 



142 



AMERICAN MACHINIST GEAR BOOK 



ANGLE 
INCREASE. 


DIAMETER 
INCREASE. 


CENTER 


CENTER-ANGLE-HUNDREDTH-DEGREES 


CENTER 


DIAMETER 
INCREASE. 


DIVIDE 


DIVIDE 


ANGLE 




ANGLE 




BY 


BY PITCH 


FOR 




FOR 


DIVIDE 
BY PITCH 


TEETH IN 


OF 


PINION 


LEFT-HAND COLUMN READ HERE 


GEAR 


OF GEAR 


PINION 


PINION 












2.00 





.00 
.0000 


.17 
.0029 


■33 


.50 
.0087 


.67 
.0116 


•83 
.0145 


I.OO 

•0175 


89 




I 


.0058 


•03 


2 


2.00 


I 


•0175 


.0204 


.0233 


.0262 


.0291 


.0320 


•0349 


88 


.07 


4 


2.00 


2 


•0349 


.0378 


.0407 


.0437 


.0466 


.0495 


.0524 


87 


.10 


6 


2.00 


3 


.0524 


•0553 


.0582 


.0612 


.0641 


.0670 


.0699 


86 


.14 


8 


1.99 


4 


.0699 


.0729 


.0758 


.0787 


.0816 


.0846 


.0875 


85 


.17 


lO 


1.99 


5 


.0875 


.0904 


•0934 


.0963 


.0992 


.1022 


.1051 


84 


.21 


12 


1.99 


6 


.1051 


.1080 


• IIIO 


•1139 


.1169 


.1198 


.1228 


83 


.24 


14 


1.98 


7 


.1228 


.1257 


.1278 


•1317 


.1346 


.1376 


.1405 


82 


.28 


i6 


1.98 


8 


.1405 


•1435 


.1465 


.1495 


.1524 


•1554 


.1584 


81 


•31 


i8 


1.98 


9 


.1584 


.1614 


.1644 


.1673 


.1703 


.1733 


.1763 


80 


•34 


20 


1.97 


10 


.1763 


•1793 


.1823 


•1853 


.1883 


.1914 


.1944 


79 


.38 


22 


1.96 


II 


.1944 


.1974 


.2004 


•2035 


.2065 


-2095 


.2126 


78 


.41 


24 


1.96 


12 


.2126 


.2156 


.2186 


.2217 


.2247 


.2278 


• 2309 


77 


•45 


26 


1-95 


13 


.2309 


•2339 


.2370 


.2401 


.2431 


.2462 


•2493 


76 


•48 


28 


1.94 


14 


•2493 


.2524 


.2555 


.2586 


.2617 


.2648 


.2679 


75 


•51 


30 


1-93 


15 


.2679 


.2711 


.2742 


.2773 


.2805 


.2836 


.2867 


74 


•55 


32 


1.92 


16 


.2867 


.2899 


.2931 


.2962 


.2994 


.3026 


•3057 


73 


•58 


34 


1.91 


17 


•3057 


.3089 


.3121 


•3153 


.3185 


•3217 


•3249 


72 


.62 


36 


1.90 


18 


•3249 


.3281 


•3314 


•3346 


•3378 


•341 1 


.3443 


71 


•65 


31 


1.89 


19 


•3443 


•3476 


.3508 


•3541 


•3574 


.3607 


.3640 


70 


.68 


39 


1.88 


20 


.3640 


■3^73 


.3706 


•3739 


•3772 


.3805 


•3839 


69 


•71 


41 


1.86 


21 


•3839 


.3872 


.3906 


•3939 


•3973 


.4006 


.4040 


68 


•75 


43 


i.8s 


22 


.4040 


.4074 


.4108 


.4142 


.4176 


.4210 


•4245 


67 


.78 


45 


1.84 


23 


.4245 


.4279 


.4314 


.4348 


.4383 


.4417 


•4452 


66 


.81 


47 


1.82 


24 


.4452 


■4487 


•4522 


•4557 


•4592 


.4628 


.4663 


65 


.84 


49 


1.81 


25 


.4663 


.4699 


.4734 


.4770 


.4806 


.4841 


•4877 


64 


.88 


50 


1.79 


26 


.4877 


•4913 


.4950 


.4986 


.5022 


•5059 


.5095 


63 


.91 


52 


1.78 


27 


•509s 


•5132 


.5169 


.5206 


•5243 


•5280 


•5317 


62 


•93 


54 


1.76 


28 


•5317 


•5354 


•5392 


.5430 


•5467 


•5505 


•5543 


61 


•97 


56 


1.74 


29 


•5543 


.5581 


.5619 


.5658 


.5696 


•5735 


•5774 


60 


I.OO 


57 


1-73 


30 


•5774 


.5812 


.5851 


.5890 


•5930 


•5969 


.6009 


59 


1.03 


59 


1. 71 


31 


.6009 


.6048 


.6088 


.6128 


.6168 


.6208 


.6249 


58 


1.05 


61 


1.69 


32 


.6249 


.6289 


•6330 


.6371 


.6412 


•6453 


.6494 


57 


1.08 


63 


1.67 


33 


.6494 


•6536 


.6577 


.6619 


.6661 


.6703 


•6745 


56 


I. II 


64 


1.65 


34 , 


•6745 


.6787 


.6830 


.6873 


.6916 


•6959 


.7002 


. 55 


1. 14 


66 


1.63 


35 


.7002 


.7046 


.7089 


•7133 


.7177 


.7221 


.7265 


54 


1. 17 


68 


1.61 


36 


•7265 


.7310 


•7355 


.7400 


.7445 


.7490 


•7536 


53 


1.20 


69 


1-59 


37 


•7536 


.7581 


.7627 


.7673 


.7720 


.7766 


•7813 


52 


1.23 


71 


1-57 


38 


.7813 


.7860 


.7907 


.7954 


.8002 


.8050 


.8098 


51 


1. 25 


72 


1-55 


39 


.8098 


.8146 


.8195 


•8243 


.8292 


.8342 


.8391 


50 


1.28 


73 


1-53 


40 


.8391 


.8441 


.8491 


.8541 


.8591 


.8642 


•8693 


49 


I-3I 


75 


I-5I 


41 


.8693 


.8744 


.8796 


.8847 


.8899 


•8952 


.9004 


48 


1-33 


77 


1.48 


42 


.9004 


•9057 


.9110 


.9163 


.9217 


.9271 


•9325 


47 


1.36 


79 


1.46 


43 


•9325 


.9380 


•9435 


.9490 


.9545 


.9601 


•9657 


46 


139 


80 


1-43 


44 


.9657 


.9713 


.9770 


.9827 


.9884 


•9942 


I.OOOO 


45 


1.41 


81 


1.41 


45 


I.OOOO 
I.OO 


1.0058 
"83 


1.0117 
.67 


1.0176 


1.0235 
■33 


1.0295 
•17 


^355 
.00 


44 


1^43 




•50 










RIGHT-HAND COLUMN READ HERE 







Table 21 — Diameters and Angles of Bevel Gear — Shaft Angles 90 Degrees 

Becker Milling Machine Company 



BEVEL GEARS 143 

MACHINING BEVEL GEARS 

Cutting bevel-gear teeth necessitates quite intricate operations if a 
correctly proportioned tooth is to be finished on other than a generating 
machine. In fact, so intricate would be the required adjustments that quite 
radical departures from a true form of tooth are frequently resorted to — 
modifications that are almost considered standard, so universally are they 
employed. 

The recognized standard tooth for bevel gears, after which all modifica- 
tions are modeled, is of the octoid form. This form of tooth, which was 
described in a preceding section, is simply a modification of the involute and 
can be accurately conjugated only by a generating machine. 

MACHINES FOE CUTTING BEVEL GEARS 

Milling machines are the ones most generally employed in cutting bevel 
gears unless extreme accuracy in the form of teeth is essential. These 
machines readily cut teeth of uniform depth of correct profile, and are also 
used for the cutting operations of bevel gears with teeth of approximately 
correct form — the type of bevels which are finished by filing the teeth to 
approximately correct profile. 

Planing machines are employed in similar operations and also for cutting 
teeth which closely resemble the true octoid form. Somewhat special 
planers, known as bevel-gear planing machines, are employed for this more 
exacting work. Three templets are used which guide the simple planing 
tools: A straight-faced one for gashing, and two formed ones, one for each 
side of the tooth space. All the tooth spaces are first gashed, one side of 
each tooth finished, and the gear completed by finishing up the other side of 
the teeth. 

Generating machines are the third type of machines used for cutting the 
teeth of bevel gears. These machines all use the generating principle, but 
either the shaping or milling process may be employed. The generating 
machines operating on the shaping process have a crown gear or its equivalent 
provided with cutting tools representing its sides in successive positions. 
Meshing with this is a large master bevel gear carrying on its arbor the gear 
to be cut. The master and crown gears rotate in mesh, thus presenting the 
work to the cutting tools in successive positions. The blank and the cutting 
tools virtually roll together to conjugate the teeth. Generating machines 
employing the milling process differ in having the sides of the teeth of the 
imaginary crown gear represented by the plane faces of milling cutters. 
The crown gear equivalent is stationary and the master bevel gear is rolled 
over it, thus presenting the work to the cutting tools. The action in the two 
machines is identical, the conjugation of true octoid teeth. 

Of these types of machines for cutting bevel gears, the milling machine is 
generally employed on account of the speed with which the work can be 



144 AMERICAN MACHINIST GEAR BOOK 

performed. Nevertheless the operations required are quite intricate and 
deserve special attention. The meager description of the bevel-gear planer, 
on the other hand, should suffice to explain its operations and, as generating 
machines are fully automatic, Httle explanation of their action is necessary. 

MILLING BEVEL GEARS 

Though rotary cutters do not lend themselves to the production of really 
high-class bevel gears, in view of the many conditions governing the shape of 
the correctly proportioned tooth, yet gears cut on milling machines are 
nevertheless entirely satisfactory for many purposes, and until the time 
arrives when all gear teeth are really standardized, will continue to be used 
quite extensively. 




a 

PIG. lOO. PARALLEL DEPTH BEVEL GEARS. 



The fact that the depth of the true octoid tooth decreases, as well as the 
outside and pitch diameters of a bevel gear, as the cone apex of the gear is 
approached, makes sole reliance upon a rotary cutter impossible. Notwith- 
standing a correct profile to the cutter, the tooth must be gone over with a 
file to remove surplus material, etc. This drawback led to the development 
of the Pentz Parallel Depth Gear in which the depth of the tooth is constant 
throughout its entire length. See Fig. loo. 

By this radical departure from the true octoid, the necessity of finishing 
tne teeth with a file is avoided — the form of tooth being correct at one point, 
it is correct at every other. This form of tooth has one possible drawback, 
however, although it is quite generally considered as the equal of the true 
octoid tooth of varying depth. 

The cutter for the parallel depth tooth is selected for the inner or smaller 
pitch circle, so there is always a certain deficiency in the amount of metal at 
the outer ends of the teeth, just as there is a surplus of metal at the inner ends 
of the teeth which has to be filed away when the cutter is selected for the 



BEVEL GEARS 



I4S 



outer pitch circle — the method when milHng the standard varying depth 
tooth. 

The Pentz Parallel Depth bevel gears are quite distinctive, owing to the 
stubby appearance of the teeth at the outer edge of the gear, but as this form 
of tooth is somewhat easier to mill than the one in which the depth of tooth 




PIG. lOI. FIRST CUT IN MILLING BEVEL GEARS. 

varies, a detailed explanation of the steps required in machining such a gear 
will be more readily grasped than if the more compHcated method necessitat- 
ing more adjustments be first considered. The more involved method can 
then be quite easily comprehended. 

The pitch of the cutter is determined from the small end of the gear, and 
the form of the cutter from the average back cone distance, as at h-h (Fig. 
I go). This will give the average form of tooth, as it is apparent that the true 

10 



146 AMERICAN MACHINIST GEAR BOOK 

form cannot be maintained the entire length of face. The only change in 
form is due to the reduced back cone distance from a to c; there is no change 
due to reducing the depth of cut, as is found in the ordinary methods of 
bevel gear milling, the pitch line of the bevel gear being cut, and the pitch 
line of the cutter coinciding during the entire cutting operation. 

The milling of a parallel or, for that matter, tapered tooth bevel gears 
will be better understood if the pitch is considered at the small end of the 
tooth. When taking side cuts, this pitch alone should be kept in mind and 
the matter will appear in a new light. 

After the first central cut has been taken as illustrated by Fig. loi the 
blank is rolled to the right, bringing the pitch line of the left-hand side of 
space parallel with the travel of cutter, as shown by Fig. 102. This move- 
ment is accomplished by indexing the blank one-quarter as many holes in the 
index plate as are used altogether to space the teeth. The table is then 
moved toward the nose of the spindle a distance equal to one-half the tooth 
thickness at small end, or one-quarter the pitch at small end. This will 
bring the pitch line of gear to the pitch line of the cutter, and the blank in 
position for first side cut, as shown in Fig. 103. 

After this cut has been made, roll the gear blank to the left, one-half as 
many holes in index plate as are used altogether to space the tooth, as per Fig. 
104, and move the table away from the nose of spindle the thickness of tooth 
at small end; this will bring the other side of the tooth into the same relative 
position, and the blank is in position for the second side cut, as in Fig. 105. 

It will be noted that the pitch is figured at the small end of the tooth, an 
ordinary spur gear cutter corresponding to the pitch at this point being used. 

The half tone. Fig. 106, shows a pair of parallel tooth gears, sent with W. 
Allen's original article. Their operation was entirely satisfactory and the 
teeth gave no evidence of being filed. These samples were 25 teeth, 10 pitch 
at small end and i-inch face. No. 2, 10 pitch cutter used. 

Referring to Figs. loi to 105; in moving the table forward and back one- 
half the thickness of space at the small end there is a smail error due to the 
fact that instead of these moves being made in the direction of the pitch line 
they are made tangent to it, as illustrated by Fig. 107, which is exaggerated 
to show this clearly, c representing the distance actually moved and / the 
theoretical distance. This error, however, is on the safe side, so that setting 
the machine as directed will allow a little clearance, depending, of course, 
upon the diameter and pitch of gear, but will not be noticed except in extreme 
cases. 

In milling the taper-tooth t3rpe of bevels, very similar steps are taken, 
but, of course, the face and cutting angles are not the same. This naturally 
complicates the holding and adjusting of the work to a considerable extent. 

The cutter should be selected with a correct profile for the outer ends of 
the teeth, but its thickness must be such as to allow it to pass between the 
teeth at the inner edge of the gear. The necessary adjustments of the blank 



BEVEL GEARS 



147 



Distance to move 
Cutter 




FIG. 102. POSITION OF BLANK WHEN ROTATED ONE-QUARTER OF THE INDEX. 




FIG. 103, CUTTER IN POSITION FOR THE FIRST SIDE CUT. 




Distance to 
move Cutter 



FIG. 104. BLANK IN POSITION FOR SECOND SIDE CUT. 



148 



AMERICAN MACHINIST GEAR BOOK 




FIG. 105. CUTTER IN POSITION FOR SECOND SIDE CUT. 




FIG. 106. A PAIR OF PARALLEL DEPTH BEVEL GEARS IN MESH. 




Filed Surface 




FIG. 107. DIAGRAM SHOWING ERROR IN 
SETTING OVER CUTTER FOR SIDE CUT 
ON MILLED BEVEL GEARS. 



FIG. 108. THE FILED SURFACE OF 
MILLED BEVEL GEARS. 



BEVEL GEARS 149 

are made, usually by trial, and the cuts taken in a manner very similar to 
that described for the parallel-depth method. The teeth are milled to a cor- 
rect outer end thickness and the surplus metal toward the inner edge removed 
by a file. See Fig. 108. 

The reason that this excess metal is left by the cutter is that, though the 
set of teeth as cut are correct throughout their entire length at the pitch line, 
their curvature is correct only at the outer end of the teeth. In the octoid 
form of tooth, for which the cutters were selected, the radius of curvature 
grows less and less as the pitch diameter of the gear decreases — i.e., toward 
its inner edge. Rotary cutters, being unable to alter their curvature, leave 
a surplus of metal outside the pitch line which gradually increases in thick- 
ness as the inner edge of the gear is approached. This surplus can only be 
removed by filing. In performing such an operation, great care is necessary 
not to reduce the thickness of the teeth at the pitch line. Particularly is 
this so if the gear is one of wide face, as the greater the face of the teeth, the 
more filing required. 

THE USE or GENERATING MACHINES 

If the gears to be finished are stocked out before being mounted on the 
generating machine, these machines are really fully automatic and when 
once set up require no further attention until the job is completed. Stock- 
ing out on generating machines is not to be recommended ordinarily, although 
the larger machines are equipped for this operation. Generating machines 
are essentially finishing machines with delicate adjustment of cutting tools 
that finish a stocked out tooth in one generating operation, so that the 
rougher and heavier work of stocking out should advisably always be per- 
formed on some other and more rugged machine. This will enable the ca- 
pacity of the generating machine to be realized fully, and operations in quan- 
tities carried through rapidly and efficiently. 

EFFICIENCY OF BEVEL GEARS 

The chief cause of decreased efficiency in bevel gearing is due to inaccu- 
racies in the shape and form of the teeth, whereby a lateral thrust is produced 
which tends to force the gears out of mesh. This is particularly noticeable 
in gears finished on milling machines or on planers unless the gears have teeth 
of the parallel-depth type, owing to the fact that it is seldom that the center 
angles of the gear and pinion coincide as they roll together. The tooth 
pressure cannot be normal between all points of engaging teeth and in such 
case a decided lateral thrust tending to separate the teeth is produced. 
Furthermore, the pinion, and usually the gear also, must be overhung so 
that the thrust is greatly increased at the bearing by leverage, materially 
increasing friction, twisting the shaft, etc. 

Even if it were possible to produce a perfect tooth on a generating ma- 
chine, so that the tooth pressure is transmitted normally, inefficiencies that 



I50 AMERICAN MACHINIST GEAR BOOK 

cannot be avoided arise, due to the angularity of the gear and pinion shafts. 
The greater the center angle of a bevel gear, the greater, as a rule, the loss 
in efficiency. The most efficient center angle to employ is that of 45 degrees, 
for then both the gear and the pinion suffer equally in respect to angularity. 
Correctly proportioned teeth and adequate shaft diameters with minimum 
backing are particularly essential for the satisfactory operation of bevel 
gears. Unsatisfactory tooth speed and wide differences between the number 
of teeth on the gear and on the pinion are also more serious drawbacks in 
bevel gearing than in spur gearing, owing to the fact that a bevel gear has 
pitch circles and corresponding tooth speeds varying from that of the inner 
or smaller end of the gear through to that of the larger end. 

HARDENING BEVEL GEARS 

The abrasive wear on gear teeth is always appreciable if the profile of the 
teeth varies even slightly from the correct form, and once such deterioration 
commences, it rapidly becomes serious. This is more noticeable in bevel 
gearing than it is in spur gearing, owing to the greater difficulty of securing 
the correct tooth profile in the former. To overcome or minimize this 
destruction, case-hardening of the gears is the logical procedure. 

The great demand for satisfactory gears, bevels as well as other types of 
toothed wheels, in automobile construction has probably been the main 
reason for the knowledge gained in recent years of the art of case-hardening 
gears. The transmission of the mechanically propelled vehicle must possess 
exceptional wearing qualities to give satisfaction, and, though other intricate 
and delicate machinery also requires gears capable of resisting the abrasive 
action of teeth slipping upon one another, it is the automobile that has made 
this subject one of such general interest. 

Processes of hardening gears, of course, differ in nearly every shop, but the 
general requirements and results secured are similar. The accurately cut 
gear should first be slowly and uniformly preheated to avoid undue distortion 
and to bring the metal to as homogeneous a condition as possible. The gear 
should then be subjected to the chemical treatment by which the skin of the 
metal is prepared for the hardening quench. Finally the quenching opera- 
tion is performed, usually in an oil bath, and the gears rapidly but uniformly 
cooled. 

One process of hardening automobile gears which has proved very success- 
ful and satisfactory is to preheat the gears in a commodious gas forge or oven 
until they become dull red (at a temperature of from 1,300 to 1,400 degrees 
Fahrenheit) and then boil them in a solution of cyanide of potassium for 
about an hour. The gears are then immediately plunged into a bath of 
fish oil which is kept in constant circulation to guard against localized over- 
heating of the oil. 

Gears hardened by this process show remarkably little warping and are 
uniformly hard — the case-hardening penetrating an average depth of 0.025 



BEVEL GEARS 151 

inch — and are practically free from any change in volume or distribution 
of metal. The average shrinkage of the diameter of shaft holes, in a large 
number of automobile transmission gears so treated, is reported to have been 
only about 0.0005 inch (total shrinkage). 

GRINDING BEVEL GEARS 

Gears that have been most carefully case-hardened usually show some 
deformation, however, and no matter how slight this is their smooth and 
proper action is impossible. Thus the benefits derived from hardening are 
partially counteracted. The distortion that is noticeable in a gear that has 
been carefully hardened, however, is so slight that it can be localized in the 
hub and there corrected. That is, the gear may be mounted on a rotary 
machine so that the teeth run true and any deformity thereby concentrated 
in the hub, which may be rebored, the bore straightened and the hub cor- 
rectly faced. 

A simple method of trueing up hardened bevel gears for grinding, and hold- 
ing them in such position, was described in Ameeican Machinist, July 20, 
iQii. Briefly summarized, the salient points in this description follow: 
The gears are first clamped to a face plate by a bolt through their bore, set 
true by using a test indicator, and kept in position by four perfectly round 
and straight pins placed equidistant on the periphery of the gear and held 
in place by rubber bands. The gear is then firmly strapped to the face 
plate, the bolt through the bore removed, and the bore and back of the hub 
properly ground. 

Draw-in chucks are also used for holding gears for grinding. The gears 
are held by the roots of the teeth, by the pitch line, or by the top of the teeth. 

OTHER TYPES OF BEVEL GEARS 

The unprecedented demand for bevel gears in automotive devices has led 
to the development of a number of meritorious types of bevels and to new 
methods of gear production which have greatly added to the scope of bevel 
gearing. Among the type developments may be mentioned the Gleason 
Spiral Bevel Gear and the Williams "Master Form" Gearing which will be 
described briefly in a subsequent section. 

The Anderson process of rolling gears from heated metal blanks — Section 
XVI — has also contributed a new form of bevel gear in the Herringbone bevel. 



SECTION VI 



Worm Gears 



An interesting model of half a dozen sets of worm gears is shown in Fig. 
109. All the gears are of the same diameter with teeth of the same normal 
pitch, though the respective speed ratios of the varius sets differ. The hori- 




FIG. loy. MODEL OF SPIRAL GEARS OF VARIOUS RATIOS. 

zontal shaft in making 32 revolutions causes the vertical gear in the fore- 
ground to make but one complete revolution, the second, two complete turns, 
and each of the succeeding vertical gears twice the number of revolutions 
made by the gear immediately in front of it, the furthermost gear making 32 

152 



WORM GEARS 153 

revolutions to 32 turns of the horizontal shaft The consecutive speed 
ratios, commencing with the set in the foreground, are respectively 32 :i, 
16:1, 8:1, 4:1, 2:1 and 1:1, the revolutions of the horizontal shaft being 
named first. 

The first three drives are evidently worms of single, double and quadruple 
thread respectively, but they are also spiral gears as the driven gears are cut 
with spiral teeth, not simply hobbed as is customary in laying out worm 
gears. The balance of the drivers resemble spiral gears even more, par- 
ticularly the most remote, showing that the so-called ''worm" is in reality a 
toothed gear of the spiral type. 



NOTATION FOR WORM GEARS 

N = number of teeth in worm wheel. 

n = number of threads in worm. 
p' = circular pitch (distance from center to center of teeth). 

L = lead (advance of worm in one revolution). 
D' = pitch diameter of worm wheel. 
T = throat diameter of worm wheel. 
D = outside diameter of worm wheel. 
F = face of worm wheel. 

a = distance from center line to point of tooth. 

b = length of side. 
d' = pitch diameter of worm. 

d = outside diameter of worm. 
d" = bottom diameter of worm. 

e = radius at throat of worm wheel. 

</) = angle of sides of face. 

B = center distance. 

R = number of revolutions of worm to one of wheel. 

5 = angle of teeth in wheel with axis (used for gashing teeth). 

IT = 3.I416. 

W = working depth. 
W = whole depth. 

/ = clearance. 

/ = thickness of tooth at pitch line. 

/'' = normal thickness of tooth at pitch line. 
p'"" = normal circular pitch. 
5 = addendum. 

U = width of worm thread at top. 

Y = width of worm thread at bottom. 

p = diametral pitch. 



154 AMERICAN MACHINIST GEAR BOOK 

FORMULAS FOR WORM GEARS 

P' 
D' = A^/ 0.3183. 

r = (.v + 2) ^'0.3183. 

D=T-\-2{e — e cos. 4>). 

F = l--\- 0.17 p'\ sin. 4> , , , ^,. 

^ ^_^ d + (0.34 p') 

~ J or ^ , when = 30 degrees. 

a = F — {b sin. (/>). 
b = W' + (0.12 /). 
d^ = as small as possible. (See discussion.) 

d = d' + 2S. 

d" = d- 2 W. 

d' 
e = s. 

2 

4> = 30° to 35°, or sin. = -— — . 

d + (0.34 p') 

D' + d' 



B = 



2 



L = p'n. 

N 
n = — 
R 



Tan. 5 = — 7/ 
-nd 

/" = / COS. 8,ort = ^^^ = ^^^ when 8 = 14^ degrees. 
f = 0.1 t. (See discussion.) 

U = 0-335 P\ or ^'^P • (See discussion.) 
P 

Y = 0.3 1/>', or ^1^244. (See discussion.) 

p ^ = p COS. b, OT p' = ^ 



COS. 8 

W = 0.6866 p'. (See discussion.) 
Formulas for tooth parts as given for spur gears apply to worm gears. 

DISCUSSION OF FORMULAS 

N, D' and p' . The number of teeth, the pitch and the pitch diameter of 
worm gears are calculated in the same manner employed for spur gears. 



WORM GEARS 



155 



T. The throat diameter of a worm gear or wheel corresponds to the out- 
side diameter of a spur gear of the same number of teeth and the same pitch. 

D. The extreme outside diameter of a worm-wheel can be found by the 
formula given, but ordinarily the measurement of a carefully drawn sketch 
is sufl&ciently accurate. Insufl&cient stock for a sharp tooth edge, depicted 



^s."^ 




YIG. no. DIAGRAMS FOR 
WORM GEAR. 

on Fig. no, is preferable to enough stock for a perfect tooth as it is safer to 
handle a gear without the sharp-cornered teeth and also because such gear 
is of more pleasing appearance. 

F. There is no gain in making greater than 30 degrees, so the second 
formula for F is the one usually employed. 

d' . The angle of the worm, 5, governs in large part the efficiency of a 
worm drive, so the pitch diameter of the worm should be made as small as 

possible. When the lead is fixed, however, d' is also fixed for tan, 8 = —j,- 

(f). This angle is usually from 30 to 35 degrees, preferably 30 degrees. 
When made as great as 45 degrees and the face of the gear widened to corre- 
spond, the gear will wear out rapidly. 

L. The lead is the advance of the worm thread in one complete revolu- 
tion and is found by multiplying the circular pitch by the number of threads 
in the worm. For instance, a 1.5-inch pitch gear with double threads has 
a 3-inch lead. 

n. The number of threads required for a worm is found by dividing the 
number of teeth in the worm-wheel by the velocity ratio required. For 



156 



AMERICAN MACHINIST GEAR BOOK 



example: If a worm gear has 60 teeth and a velocity ratio of 30 to i is 
required, the worm should have two threads! — = 2J- 

/, U, Y , and W'. The formulas given for these various dimensions apply 
to worms cut to 14}^^ degrees standard. Any deviation from this standard 
obviously calls for modifications of these formulas, but no fixed rule can be 
advanced. 

The length of the worm (see Fig. in) need be no more than three times 
the circular pitch, as seldom do more than two teeth come in contact with the 



^?J at pitch 
I line 




riG. Ill, WORM. 



wheel teeth at the same time. It is good practice, however, to make the 
worm about six times as long as the circular pitch so that it may be shifted 
as it becomes worn, the worm nearly invariably wearing more rapidly than 
the wheel. 

REVERSIBLE WORM AND GEAR 

The surface of the worm thread constantly slips and slides over the sur- 
faces of the wheel teeth, the direction of slippage following the plane of con- 
tact. The slippage plane being mutually tangent to the face of the thread 
and the face of the wheel teeth, the gliding angle must equal the sum of the 
lead angle and half the angle included between the faces of the worm thread. 
When this gliding angle equals 45 degrees it is evident that it is immaterial 
whether the worm or the gear is the driver, or which is the driven member — 
the worm and the gear will be perfectly reversible in this respect. 

Expressed in the form of a simple equation, the conditions requisite for 
the worm and gear to be perfectly reversible are as follows: 

F = — = 45 degrees 



WORM GEARS 



157 



where 

Y = lead angle, 

X = angle included between the faces of the worm thread. 

This equation is generally applicable, but there are reasonable limits to 
the lead angle that can be efficiently employed. The lower limit for the lead 
angle is about 25 degrees, making necessary an included angle of 40 degrees 
in order to make the gliding angle one of 45 degrees. A more acute lead 
angle would necessitate an included angle so obtuse as not to give adequate 
contact surface. The upper limit is, of course, 45 degrees. In this case there 
could be no included angle. This condition could exist only when the profile 
of the worm thread is perpendicular to the axis of the worm, when the worm 
thread is square or rectangular in cross-sections. 



LEAD 


INCLUDED 






LEAD 


INCLUDED 


ANGLE, Y 


ANGLE, 


X 






ANGLE, Y 


ANGLE, X 


25 


40 








36 


18 


26 


38 








37 


16 


27 


36 








38 


14 


28 


34 








39 


12 


29 


32 








40 


10 


30 


30 








41 


8 


13 


28 








42 


6 


S3 


26 








43 


4 


32 


24 








44 


2 


33 


22 








45 





54 


20 




THE 


HOB 







The hob for cutting the teeth on the wheel must have an outside diameter 
equal to the outside diameter of the worm plus twice the clearance (see Fig. 



U-29V 




FIG. 112. HOB. 



112), but this should advisably be increased by 0.03 X p' to allow for wear. 
The hob should always be a little longer than the section of the worm 
having any contact with the wheel (see Fig. 113) and, as this depends upon 
the diameter of the wheel to be cut, it should be proportioned to the diameter 
of the largest wheel liable to be handled. The correct hob length may be 
found from the following simple formula: 

L = 2y/{D - W) - W 



158 



AMERICAN MACHINIST GEAR BOOK 




FIG. 113. LENGTH OF THE HOB. 



where 



L = length of hob, 

D = outside diameter of largest gear to be cut, 
W = whole depth of tooth. 

NUMBER OF FLUTES 

The following article by Oscar J. Beale, which originally appeared in 
American Machinist, June 22, 1899, very ably discusses the proper 
arrangement of teeth on a worm-gear hob. 

''In the works of the Brown & Sharpe Manufacturing Company a pair of 
gears was wanted of the spiral or screw type, and it was thought better to 




FIG. 114, WORM GEAR B AND 
WORM A. 



FIG. 118. HOB USED FOR CUTTING 
WHEEL B OF FIG. II4. 



WORM GEARS 



159 



make the large gear, or member, as a worm and the small member as a 
worm-wheel. 

Fig. 114 shows the worm and wheel in mesh; A is the worm and B is the 
worm-wheel. The large member, or the worm A, has 43 threads; the lead of 
the worm is 60.3 inches, and the thread pitch, or the axial pitch, is 1.4 inches. 
The small member, or the worm-wheel B, has 7 teeth, and the circular pitch of 
the wheel is, of course, the same as the thread pitch of the worm, 1.4 inches. 

Fig. 115 is an axial section of the worm threads. The threads incline 57 
degrees from the plane perpendicular to the axis, which is so great that, while 
the axial thickness of the thread at 
the pitch line is J^f inch, the actual 
or the normal thickness is not quite 
^0 inch. In Fig. 116 the Hne CD 
shows the inclination of the threads ; 
CE is the axial pitch, and F G the 
actual or normal pitch. 

In cases where the inclination of 
the thread is more than 15 degrees, 
that is, in cases where the normal 
pitch is less than 0.96 of the axial 

FIG. 116. AXIAL AND NORMAL PITCH. 



h-^383-^ 






FIG. 115. 



AXIAL SECTION OF WORM 
THREADS. 



FIG. 117. NORMAL SECTION 
OF WORM THREADS. 



pitch, it is well to have the depth and the addendum correspond to the nor- 
mal pitch. Fig. 117 is a normal section of the thread, the depth being the 
same as a gear tooth of equal pitch, which makes the thread look shallow 
and thick when seen in the axial section. Fig. 115. 

The worm-wheel B, Fig. 114, was hobbed, or cut, with the hob shown in 
Fig. 118. 

The worm has more than six times as many threads as the worm-wheel; 
the pitch diameter of the worm is four times that of the wheel; the wheel is the 
driver. The hob is made up of a cast-iron body, upon which are fastened lags 
that are arranged in steps in order to have the lags alike for convenience in 
manufacturing. Once a large hob was made that did not work, because the 



l6o AMERICAN MACHINIST GEAR BOOK 

cutting edges of the hob teeth did not trim the tops of the worm-wheel teeth 
narrow enough to clear the backs of the hob teeth, which jammed so hard 
that the machine could not go. This jamming of the backs of the hob teeth 
upon the tops of the worm-wheel teeth was owing in part to incorrect spacing 
of the lags HH, Fig. ii8, which will be explained. 

A worm is a screw whose threads have the same outline, upon an axial 
section, as the teeth of a rack, the purpose of a worm being to mesh with a 
gear. A worm gear meshes with a worm. The action of a worm meshing 
with a worm gear is analogous to that of a rack with a spur gear, as stated 
by Professor Willis in his "Principles of Mechanism." In most worms the 
outlines of the threads upon the axial section have straight sides, as in Fig. 
115, which corresponds to the sides of rack teeth in the involute system of 
gearing. 

Fig. 118 is a hob made up of cast-iron body, into which are fastened lags 
HH. Two of these lags are shown detached. The lags were threaded in 
axial section like A, Fig. 115. The resulting teeth were trimmed and backed 
off, as in the detached lag on the left. The numbers in the scale are for 
inches. 

I have spoken of the failure of a hob because the backs of the hob teeth 
jammed upon the tops of the wheel teeth. This interfering action can be 
explained in several ways; it is analogous to trying to thread a coarse screw in 
a lathe with a tool that does not lead or incline in the same direction that the 
thread inclines. A thread tool inclined for a right-hand thread would soon 
interfere in cutting a left-hand thread. Any grooving tool that has only one 
cutting edge or face must track in the same groove that it cuts. Sometimes 
a tool goes wrong and cuts a groove that bends the tool, which is occasionally 
noticed in cutting off a large piece in a lathe. In cutting a deep narrow 
groove a thin saw sometimes runs so much to one side that the saw is broken. 
In the case of the hob the interfering teeth would neither bend nor break, and 
so the machine had to stop. The teeth of a hob should be so arranged that there 
will be a cutting edge to take off an interfering point as it comes in the way. 
A worm-wheel can be cut with a tool that has only one cutting edge by 
bringing the tool into different positions in relation to the teeth of the wheel. 
In the American Machinist for May 27, 1897, reference was made to the 
great number of cutting edges that a hob must have in order to cut a perfect 
wheel, and a description was given of a machine that cuts worm-wheels with 
a single tool acting in different positions. Such a machine was patented 
November 15, 1887, and another July 5, 1898. 

In most hobs the cutting edges are straight, and in consequence the sides 
of the hobbed worm-wheel teeth are made up of straight lines in warped 
surfaces that meet in angles. These angles are often not noticed in worm- 
wheels of fine pitch and in wheels having a large number of teeth; but in 
wheels of coarse pitch and in wheels having few teeth the angles may be quite 
pronounced. Fig. 119 shows a worm-wheel that has teeth with hobbly sides 



WORM GEARS 



i6i 



on account of these angles. This wheel was cut with the hob shown in Fig. 
1 20. The length and the diameter of the blank were great enough to extend 
beyond the teeth left by the hob that are available to work in connection with 
the worm. The available part of the teeth occupy about two-thirds the 
length of the wheel through the midpart, as between / and /. Though the 
teeth are available, yet their sides are so hobbly between I and / that they 
will need to have the angles finished off before the wheel can run smoothly 
with its worm. 

Another kind of stepped action of the hob is seen as grooves near KL^ 
Fig. 119, which are cut in consequence of the quick travel of the large 
part KLy in proportion to the nar- 
row flats MM, Fig. 120, at the tops 
of the hob teeth. If the travel of 
KL had been slow enough or if 





FIG. 119. A WORM GEAR OF FEW TEETH 
A>rD COARSE PITCH. 



FIG. 120. HOB USED IN CUTTING WORM 
GEAR OF FIG. 1 1 9. 



the flats MM had been wide enough, there would have been no grooves. 

The circular pitch and the number of teeth of Fig. 119 are the same as in 
B, Fig. 114. 

It is well known that the cutting edges of a hob must act upon the worm- 
wheel teeth in different positions, and that a tool with a single cutting edge 
must track in a groove cut with itself; but it was a surprise to learn that a 
hob of any number of cutting edges can be so made that it will absolutely 
refuse to cut a wheel that has only a few teeth like B, Fig. 114. When the 
workman told me that the hob jammed, I was incredulous, but a glance at 
the work proved that he was right. I could not believe that my previous 
experience had been such that I could have known how to make the hob, yet 
in a few minutes, when the solution came to me, I had the feeling that I must 
have known it well some time in the long past. 

The things that affect this interference might be called variable; there are 



11 



l62 



AMERICAN MACHINIST GEAR BOOK 



several of these variables. I am unable to give a rule that will indicate the 
conditions in which interference would be objectionable; yet, while limits may 
not be easily defined, an understanding of a few extremes may enable us to 
keep away outside these limits. 

One way of explaining the interference is based upon the fact that, in a 
gear, any point outside the pitch circle moves through a greater distance, or 
faster, than a point in the pitch circle. Fig. 121 shows a single- threaded hob 
having only one row of cutting edges 00, the teeth, or the threads, extending 
nearly around the hob. Let the teeth in the wheel P be shaped as if they had 
been cut to the full depth with the cutting edges of the hob, and in a low- 




PIG. 121. A SINGLE-THREADED HOB WITH ONE ROW OF CUTTING EDGE. 



numbered wheel we shall have tooth faces shaped as shown. Now, place this 
gear in mesh with the hob at the cutting edges, turn the hob in the direction 
of the arrow, and we shall soon find that the tooth faces of the wheel will 
interfere with the hob threads, as shown at NN, in consequence of the faces 
NN moving at a different speed from the pitch circle. There would also be 
interference upon the flanks of the gear, but it was thought that the cat would 
be quite as clear if the showing of this flank interference were not attempted. 
Only a slice section of the wheel is shown at P; in the real wheel we should 
have a still greater interference at the outer part of the teeth TT. In moving 
along a straight path, from i? to 5, a close-fitting tooth Q would not interfere; 
but interference would begin as soon as Q was moved in a curved path like 
that of a gear tooth. 

From this consideration of Fig. 121 we should conclude that it is impracti- 
cal to hob a wheel of few teeth with a hob having only one row of cutting 
edges, like the one shown. Even though we reduce the hob threads back of 
the cutting edges enough to clear the teeth of the wheel, so that it will be 
possible to hob the wheel, we shall not shape the teeth so that they will run 
correctly with the worm. 

Another illustration of interference may be seen in Fig. 122. Let a small 



WORM GEARS 



163 



gear be cut, as shown, with a cutter that is shaped like a gear, as might be 
done in a Fellows gear shaper. Let every rotative movement of the gear, in 
order to take another cut, be through exactly one tooth, a cutter tooth always 
cutting on the hne of centers, as shown. In this way cut to the full depth, 
moving the gear exactly one tooth at every setting. In our experimental 
cutting we can let the cutter rotate through one tooth at every movement of 
the gear, or we can let the cutter remain stationary, so far as rotation is 
concerned. When we have cut a few spaces to the full depth, we shall find 
that they are shaped as shown in Fig. 122, the spaces below the pitch line 

merely fitting a cutter tooth upon 
the line of centers without any envel- 
oping or shaping of the gear teeth, as 
there would be in the ordinary work- 
ing of the Fellows gear shaper. Now 
let us stop the cutter, leaving its cut- 
ting edges just above the side of the 





122. ABSENCE OF ENVELOPING 
ACTION IN GEAR CUTTING. 



T U. 

riG. 123. A DOUBLE-THREAD HOB WITH 
TWO ROWS OF CUTTING EDGES. 



gear, and try to rotate the gear with the cutter in mesh, just as if they were 
a pair of gears, and we shall at once see that the teeth of the cutter inter- 
fere back of the cutting edges, as we should suppose from a mere inspection 
of Fig. 122. 

The same kind of interference that we saw in a single- threaded hob. Fig. 
121, will occur in a double- threaded hob that has only two rows of cutting 
edges if they are evenly spaced and are parallel to the axis. This can be 
understood from Fig. 123. Any tooth or thread U is exactly opposite another 
tooth w, because the thread is double, one thread starting at the end half way 
around from the other thread. One row of cutting edges UV will pass 
through the spaces cut by the other row uv in the same position as regards 
the worm-wheel teeth, and in consequence the backs of the teeth in both 
rows will interfere, as in Fig. 121. 

A three-threaded hob with three evenly spaced rows of cutting edges will 
interfere, and so on. 



1 64 AMERICAN MACHINIST GEAR BOOK 

From a careful consideration of the foregoing we arrive at the general 
principle — The spacing of a hob must not be equal to the circumferential distance 
occupied by either one or to any whole number of threads. 

The more teeth there are in a worm-wheel the more teeth it is possible to 
have in contact with the worm threads at one time, in a worm that is long 
enough, and in consequence a long hob can possibly cut upon enough teeth 
at a time; or, what is the same thing, it can cut every tooth in enough posi- 
tions so that even with only one row of cutting edges it can shape the teeth 
smooth and without interfering. In practice, however, it is never safe to 
trust to only one row. 

My hob has 43 threads and 21 lags or cutting rows. It had spaced the 
lags ^:43 of the circumference apart, which gave just two thread spaces to each 
lag. Hence, so far as the shape of the worm-wheel teeth is concerned I was 
not doing any different with the 21 lags (shown in Fig. 118) than I could have 
done with only one lag. 

Another body was made for the hob. The lags were spaced evenly, 21 
in the circumference, which gave 2^-21 thread spaces to each lag. This ar- 
rangement afforded twenty-one positions of lags. To accommodate these 
positions steps were provided, as seen at H. The hob was successful. 

RELIEVING A SPIRAL FLUTED HOB WITHOUT SPECIAL FIXTURES* 

Special fixtures are not necessary to relieve the teeth in a spiral fluted hob. 
This may be accomplished by indexing for a greater number of flutes than 
are actually contained in the hob. 

Let L = lead of hob. 

Li = lead of flute milled in hob. 
C = pitch circumference. 

/ = distance gained by spiral flute in one revolution. 
C = circumferential length of each flute. 
N = number of flutes to be added. 



Li 


=f- 


I 


-&■ 


N 


/ 



If N turns out an inconvenient figure it may be changed to the nearest 
whole or fractional number and the lead of flute {L') changed to suit as 
follows: 

L ^ L 



* 



R. J. Briney. 



WORM GEARS 



i6s 



Example t 

What will be the proper index for the relieving attachment for a hob 4 
inches pitch diameter and 8-inch lead, number of flute cut in hob 5? 

C = TT 4 = 12.5664. 
L' = Y^ — — o — X 12.5664 = 19.739 inches. 

Li O 

Li 19-739 



N = -^, = 12.5664 = 2 — . 
C — ^ 2 c; 

Substituting 25^:5 for 2 makes our index 5 + 2 = 7, instead of 5. 
Since the value N is changed from 2^^^ to 2, we must change the lead of 
flutes to correspond. 

^ 12.5664 

Li = -^^L = 2 X ^^-^^^^ X 8 = 20 inches. 



REDUCING THE DIAMETER OF WORM GEARS 

Increasing the pitch diameter in order to avoid undercut is not good 
practice, as it tends to shorten the life of a gear, instead of lengthening it By 
referring to Fig. 124 it is plain that the I 

pitch of a worm gear at C is greater 
than at A and, since this pitch the worm 
can only be made to correspond with the 
pitch of the gear at one point, generally 
A, there must necessarily be a great 
amount of friction, with the necessary 
loss in efficiency at B and still more at 
C. 

The efficiency and life of worm gears 
are greatly increased, therefore, by 
making the diameter and, therefore, the 
pitch of the gear to correspond with the 
pitch of the worm at point B, or the 
medium pitch diameter of gear. This 
will reduce the pitch diameter the fol- 
lowing amount, it being assumed that 
angle of face is 30 degrees; or it can 
readily be found by a careful layout. 

Corrected pitch diameter of worm gear = D' — 2{d' — d' 0.97). 

There are in reality as many different pitch diameters between A and C as 
we would care to take sections, as the pitch is changing constantly. For our 
illustration, however, but the three main points have been considered. 




FIG. 124. 



CORRECTED DIAMETER OF WORM 
GEAR. 



i66 



AMERICAN MACHINIST GEAR BOOK 



GENERAL MANUFACTURING PROCESSES 

The worm gear is first gashed out by a cutter, approximating the outside 
diameter of the hob for about two-thirds the full depth of the finished tooth, 
or else a taper hob of slightly smaller angle than the worm is used for roughing 
out the worm-wheel teeth. The finishing hob is then placed on the cutter 
spindle and dropped as far as possible into the gashed out tooth. The hob 
then completes the gear, driving it around and finishing the teeth at the same 
time. In starting the finishing out, care must be taken to prevent the teeth 




Angle of 
milling 
machine 



table 



FIG. 125. CUTTING WORM GEAR WITH A 
HOB OF A DIFFERENT ANGLE FROM 
THE ENGAGING WORM. 



of the hob locking on some sharp corner left by the gashing cutter. Gears 
which have been roughly cut with taper hobs avoid this danger to a great 
extent. 

It is always advisable when bobbing the gear wheel teeth to take a hob 
that is similar to the worm to be employed, but when such a hob is not avail- 
able one that is somewhat larger or smaller than the worm can frequently 
be used. See Fig. 125. By offsetting the axis of the hob, as shown in the 
figure, correct teeth can frequently be cut — in fact, it is sometimes possible 
to cut a right-hand wheel with a left-hand hob or vice versa by swinging the 
gear around until the angle of its thread corresponds with the angle of the hob. 

STRAIGHT-CUT WORM GEARS 



A modification of the ordinary type of worm and gear has been success- 
fully employed in which the gear teeth are cut in a straight path, Hke a spur 
gear. See Fig. 126. 

Advantages are claimed for this construction. It permits side adjust- 
ment which is impossible with the ordinary type of worm gear, and the con- 
tact is believed to be better as the pitch of the wheel corresponds with that of 
the worm the full width of the face. A disadvantage of the construction 



WORM GEARS 



167 



is that the hehx angle of the worm that can be used with such a gear is limited 
to one of about 15 degrees. 

This construction is often used for elevator service as it avoids a certain 
amount of the vibration that is practically unavoidable when employing 




FIG. 126. STRAIGHT-CUT WORM GEAR. 

gears of the hobbed type. Spacing errors are easier to avoid when cutting 
the plain straight teeth. 

The teeth of the straight-cut worm gears are cut on milling machines, 
either of the plain or universal type. The table travels at right angles to 




FIG. 127. CUTTING STRAIGHT-CUT WORM GEARS ON MILLING 

MACHINE. 

the line of the cutter, the work being set up at the angle of the teeth in the 
worm gear. See Fig. 127. 

Gears of this type should be laid out like spiral gears so that a standard 
spur-gear cutter may be employed to cut the teeth. 



MATERIALS 



The constant sliding between the thread surface of the worm and the face 
of the gear teeth limits and controls the materials or combinations of mate- 
rials of which the worm and gear may be constructed. The worm being the 
active member as far as movement is concerned when slipping past the gear 



i68 AMERICAN MACHINIST GEAR BOOK 

teeth, its thread is subject to constant and more destructive abrasion than 
are the teeth of the gear. The gear teeth though not subject to constant 
wear, coming into rubbing contact with the worm but occasionally, must 
nevertheless possess good wear-resisting qualities without being so hard as 
to increase unduly the wear on the worm thread. This relationship between 
the wearing qualities of the worm and of the gear is of the utmost importance. 

It is generally recognized that the best materials from which a worm and 
a worm gear can be constructed, in order to realize good wearing qualities, 
high efficiency, etc., are case-hardened steel for the worm and phosphor- 
bronze for the gear. This combination gives excellent results. 

Attempts have been made to substitute manganese-bronze as a material 
from which to make the gear teeth — usually only the teeth and rim of a 
worm gear are constructed of bronze, the hub, arms, etc., being made of cast 
iron or other cheaper material — but with disappointing results. The man- 
ganese-bronze was unsuitable on account of its hardness. 

An excellent phosphor-bronze to employ for worm gears consists of: 
Copper, 80 parts; phosphorus, i part; tin, 10 parts. 

The steel for the worm may be almost any low-carbon steel, which can 
be readily case-hardened and does not contain more than 3 or 3.5 per cent, 
of nickel nor more than 0.16 to 0.18 per cent, carbon. In case-hardening such 
a steel it should be carried to such a point that the scleroscope indicates a 
hardness of from 60 to 70. 

The hardened steel worm should be carefully ground and trued up, but 
such operations are simple and evident for any standard type of worm. 

POWER AND EFFICIENCY OF WORM GEARING* 

In view of the good results now being obtained with worm gearing the old 
prejudice against that form of gearing, on account of its supposed low effi- 
ciency and short life, is dying out. These good results are the outcome of the 
application of principles which are by no means a late discovery, and it is 
expected that what follows will contain much that to some readers is not new. 
At the same time it is an undoubted fact that the best practice with worms is 
understood by but few, relatively speaking, and the corroboration of the theory 
by examples from practice which follow is believed to be new. No better 
illustration of the fact that good practice with worm gearing is not yet widely 
understood could be given than the statement in a recent and excellent work 
on gearing that ''the diameter of the worm is commonly made equal to four or 
five times the circular pitch," the fact being that such proportions are dis- 
tinctly bad if the worm is to do hard work. 

It should be stated at the beginning that while what follows is not offered 
as a presentation of all the data necessary for assured success with worms 
under all conditions, it is hoped to make the general conditions of successful 
practice plain, and to present the ''state of the art" as it exists to-day, 
* F. A. Halsey, in the American Machinist. 



WORM GEARS 



169 



The essential change in practice which has improved the results obtained 
with worm gearing has been an increase in the pitch angle over what was for- 
merly considered proper. There is no doubt whatever that this change has 
increased the efficiency of the gear, and, what is of more importance, has 
reduced the tendency to heat and wear. This is not only a fact, but it is a 
sound conclusion from theoretical considerations, which might have been 
predicted under proper examination. 



THEORY OF WORM EFFICIENCY 

The reason why an increase of pitch, other things being equal, or, in other 
words, an increase of the angle of the thread, gives these results, will be under- 
stood from Fig. 128. li a b he the axis of the worm and c d sl line represent- 
ing a thread, against which a tooth of 
the wheel bears, it will be seen that if the 
tooth bears upon the thread by a pressure 
P, that pressure may be resolved into two 
components, one of which, e f, is perpen- 
dicular, while the other, e g, is parallel to 
the thread surface. The perpendicular 
component produces friction between the 
tooth and the thread. The useful work 
done during a revolution of the thread is 
the product of the load P and the lead of 
the worm, while the work lost in friction 
is the product of the perpendicular pres- 
sure e f, the coefficient of friction and the 
distance traversed in a revolution, which 
is the length of one turn of the thread. 
Now, if the angle of the thread be doubled, 
as indicated, the load P remaining the 

same, the new perpendicular component J h oi P will be slightly reduced 
from the old value e J, while the length of a turn of the thread will be slighty 
increased. Consequently their product and the lost work of friction per 
revolution will not be much changed. The useful work per revolution will, 
however, be doubled, because, the pitch being doubled, the distance traveled 
by P in one revolution will be doubled. For a given amount of useful work 
the amount of work lost is therefore reduced by the increase in the thread angle, 
and since the tendency to heat and wear is the immediate result of 
the lost work, it follows that that tendency is reduced. For small angles of 
thread the change is very rapid, and continues, though in diminishing degree, 
until the angle reaches a value not far from 45 degrees, when the conditions 
change and the lost work increases faster than the useful work, an increase of 
the angle of the thread beyond that point reducing the efficiency. 




FIG. 12 



THE PRINCIPLE OF WORM 
EFFICIENCY. 



lyo AMERICAN MACHINIST GEAR 1300K 

This general consideration of the subject shows the principles at the 
bottom of successful worm design, but a more exact examination is desirable. 
According to Professor Barr the efficiency of a worm gear, the friction of the 
step being neglected, is: 

_ tan. a (i — f tan. a) 

tan. a +/ 
in which 

e = efficiency, 

a = angle of thread, being the angle dfioi Fig. 128, 

/ = coefficient of friction. 

To study the effect of the step, a convenient assumption is that the mean 
friction radius of the step is equal to that of the worm. This assumption 
would be realized only in cases where the step is a collar bearing outside the 
worm shaft, and the preceding and following formulas therefore represent 
extreme cases, one of a frictionless step, which would be approximated by a 
ball bearing, and the other of a step having about the extreme friction to be 
met with. Most actual cases would therefore fall between the two. Again, 
according to Professor Barr, the efficiency of a worm and step on the above 
assumption is:* 

tan. a (i — ftan. a) , . , . 

e = 7 — (approximately). 

tan. a + 2 / 

Notation as before. 

These formulas give no clear indication of the manner in which the effi- 
ciency varies with the angle, and Chart 10 has been constructed to show this 
to the eye. The scale at the bottom gives the angles of the thread from o to 90 
degrees, while the vertical scale gives the calculated efficiencies, the values 
of which have been obtained from the equations and plotted on the diagram. 
The upper curve is from the first equation, and gives the efficiencies of the 
worm thread only; while the lower curve, from the second equation, gives the 
combined efficiency of the worm and step. In the calculations for the dia- 
gram it is necessary to assume a value for/, and this has been taken at 0.05, 
which is probably a fair mean value. The experiments made by Mr. Wilfred 
Lewis for Wm. Sellers & Co. showed an increase of efficiency with the speed. 
The present diagram may be considered as confined to a single speed, and 
at the same time is not to be understood as showing the exact efficiency to be 
expected from worms, but rather to exhibit to the eye the general law con- 
necting the angle of the thread with the efficiency. 

The curves will be seen to rise to a maximum and then to drop. The exact 
values of the angle of thread to give maximum efficiency may be easily found 
by the methods of the calculus, the results being: 

For worm thread alone the efficiency is at a maximum when 

tan. a = \i +P - f. 

* In Professor Barr's formulas it is assumed that the worm thread is square in section. 
Thread profiles in common use affect the result but little. 



WORM GEARS 



171 



Efficiency Percent 



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172 AMERICAN MACHINIST GEAR BOOK 

Substituting the value of / (0.05) used in calculating the diagram, this 
becomes tan. a for maximum efficiency = 0.9512, and by referring to a table 
of natural tangents we find that a for maximum efficiency = 43° 34°. 
Similarly for the worm and step the result is tan. a for maximum effi- 
ciency = 

\2 +/4^ — 2/, which for / = 0.05 = 1.3 18, 

and a table of tangents tells us again that a for maximum efficiency = 52^49'. 

Of more importance than the angle of maximum efficiency is the general 
character of the curves, of which the most pronounced peculiarity is the 
extreme flatness, showing that for a wide range of angles the efficiency varies 
but little. Thus, for the upper curve there is scarcely any choice between 30 
and 60 degrees of angle, and but little drop at 20 degrees. 

At first sight the lower curve might be thought the more useful of the two, 
as it includes the effect of the step, but a little consideration will show that 
this is not the case. For most cases in which worms are used the efficiency 
of the transmission, as such, is of very little account. What the designer 
concerns himself with is the question of durability and satisfactory working, 
and the results to be expected in this respect are best shown by the upper 
curve, in which high efficiency means a durable worm. Throughout this 
discussion, in fact, the chief significance of efficiency lies in the fact that low 
efficiency means rapid wear, and vice versa. 

EXPEEIMENTAL CORROBORATION OF THE THEORY 

The experiments of Wm. Sellers & Co., before referred to, go far to confirm 
the soundness of the above views. From the present standpoint it is unfortu- 
nate that those experiments did not cover a wider range of worm-thread 
angles — those actually used being 5 degrees, 7 degrees, and 10 degrees. 
Other experiments were, however, made on spiral pinions of higher angles, 
spiral pinions being understood by Mr. Lewis to mean those pinions having 
the mating gear a true spur, the pinion shaft being at a suitable angle with the 
gear shaft to bring the pinion in proper mesh — a construction which is 
exemplified in the well-known Sellers planer drive. Mr. Lewis gives a 
formula by which the efficiencies of worms can be calculated from those for 
spiral pinions, and in the absence of direct experiments on worms of high 
angles, his results for spiral pinions have been modified by this formula to 
read for worms. The results for the two forms of gearing differ by less than 
5 per cent, for the extreme case of his experiments To compare the 
results obtained by Mr. Lewis with Professor Barr's formula, a speed has been 
selected from the experiments giving the nearest coefficient of friction to that 
used in obtaining the curves of Chart 1 1 . The results have been plotted in 
Chart II, where they appear as small crosses, and will be seen to have a very 
satisfactory agreement with the lower curve, with which they should be 



WORM GEARS 



173 



compared, as the steps of the worms used by Mr. Lewis were of the usual 
pattern without balls. 

The variation of the coefficient of friction with the speed lends an interest 
to Chart 11, which is a series of curves obtained from the results published by 
Mr. Lewis in the same manner as the crosses of Chart 10, the curve for 20 
feet velocity being in fact the same as that appearing as crosses on Chart 11. 
The other curves of Chart 11 are obtained from those of Mr. Lewis, and cover 
a range of velocities from 3 to 200 feet per minute at the pitch Une, as noted at 



100 



90 



80 



§ 70 

Ah 



"60 



50 



40 



30 



CHART II. 



10 20 30 

Angle of Thread Degrees 



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45 



RELATION BETWEEN THREAD ANGLE, SPEED AND EFFICIENCY WITH CASES 
FROM PRACTICE. 



the right. In this diagram the results obtained by Mr. Lewis on worms are 
plotted direct, but the experiments on spiral pinions have been modified as 
explained above. Inspection of the curves shows that while there is a pro- 
gressive increase of efiiciency with the speed, there is, nevertheless, not much 
probability, or indeed room, for further improvement beyond the speed of 
200 feet per minute. It will furthermore be seen that the efficiency drops off 
•much less for low angles of thread at high speeds than at low. 

In interpreting this diagram, it should be remembered that the durabihty 
of a worm depends upon the amount of power lost in wear, and not upon the 
percentage so lost. The ability of a given worm to absorb and carry off the 
heat due to friction is fixed, and does not vary with the speed. That is, a 
given worm running at 100 revolutions under a given pressure can carry off as 
much friction heat as the same worm at 200 revolutions, while it, under the 



174 AMERICAN MACHINIST GEAR BOOK 

same pressure, would transmit but one-half the power in the former case that 
it would in the latter. In other words, the percentage of lost work might be 
twice as much at the lower speed as at the higher without increasing the 
tendency to heat. 

The increase of efficiency with the speed is a valuable property of worms, 
and enables them to do much more work than they otherwise would. Thus 
the 2o-degree worm at 20 feet per minute lost 21 3-^ per cent, of the work in 
friction. Increasing the speed to 40 feet doubled the work applied, and, 
had the efficiency remained constant, would have doubled the friction heat to 
be dissipated. In point of fact, this increase of speed diminished the per- 
centage of loss to* 17, and the amount of loss and heat, instead of being 
doubled, was only increased in the ratio of 160 to 100. It is plain from the 
diagram, however, that this action does not continue much beyond a velocity 
of 200 feet per minute, beyond which the amount of loss must be more 
nearly proportional to the speed, and this doubtless has some connection with 
the fact observed by Mr. Lewis that 300 feet per minute is the limit of speed 
when the gears are loaded to their working strength, and that the best condi- 
tions are obtained at about 200 feet per minute. It is proper to add, how- 
ever, that in the cases from practice given later there are three which have 
been made repeatedly, and which are conspicuously successful, in which the 
velocity exceeds 600 feet, and one in which it exceeds 800 feet. No doubt, in 
all such cases, if the pressure on the teeth could be known it would be found 
to be light. 

It will be seen that an increase of speed for any worm under constant pres- 
sure leads to an increase of friction work, and the limit is reached when the 
worm is no longer able to carry off the heat generated fast enough to prevent 
undue rise in temperature. Furthermore, this limiting speed depends upon 
the pressure, it being higher for low pressures than for high. A worm having 
an angle which might be successful at low speed may fail at high speed; but it 
would seem that any worm which is successful at high speed should also be 
successful at low, which is in accordance with mechanical instinct. 

There are, it will be observed, two methods of increasing the pitch angle. 
The diameter may be kept constant and the pitch be increased, or the pitch 
may be kept constant and the diameter be reduced. From a mathematical 
standpoint, these two methods are identical; that is, at a given pitch-line 
velocity a worm of a given angle should have the same efficiency, regardless 
of the diameter; but in a mechanical sense the methods are not identical. 
The worm of the larger diameter would naturally have a gear of wider face, 
and the pair, having greater area of tooth surface in contact, would carry a 
larger load. 

EXAMPLES FROM PRACTICE 

It is impossible to say who was the first to recognize the significance of the 
pitch angle as a factor in the satisfactory performance of worm gearing, but it 



WORM GEARS 175 

may be mentioned as a matter of interest that the exhibit of the Hewes & 
PhiUips Iron Works at the Newark Industrial Exhibition of 1873 included 
several worm-driven planers, in which the worms were double- threaded and 
had a pitch angle of 15° 15', a pitch diameter of 33^2 inches, a lead of 3 inches, 
and a speed cutting of 256 and backing of 640 r.p.m., which give pitch-line 
velocities of 237 and 590 feet. This worm was successful, and was many 
times repeated; but later on Hewes & PhilHps were struck by the high-belt 
speed idea, and in order to increase the belt speed they changed the worm to 
6.16 p. d., I ^-inches pitch, single thread; speed cutting, 446, and backing 
1,110 r. p. m., giving a pitch angle of 5° 15' and pitch-line velocities of 720 
and 1,780 feet. This worm was a failure, and was soon changed to 6.16 p. d., 
3)'^-inches lead, double-thread, speed cutting, 281, and backing 700 r. p. m., 
giving an angle of 10° 15' and pitch-line velocities of 452 and 1,130 feet. This 
worm did better than the last, but not so well as the first. By this time the 
lesson was learned, and Hewes & Phillips set out to use a worm of 30 degrees 
pitch angle. Structural considerations, however, prevented the use of so 
high an angle and they compromised on 20 degrees, the final worm result- 
ing from this experience having a pitch diameter of 2.63 inches, with 3 
inches lead, quadruple thread, the speed cutting being 300 and backing 
700 r. p. m., giving pitch-line velocities of 205 and 480 feet, and this remained 
the standard angle as long as these planers were manufactured. 

The writer has seen one of these 20-degree worm gears, opened up after 
twelve years' use, and the wear disclosed was very slight — no shoulder being 
in existence. As a result of the experience outlined above, this house adopted 
the standard practice of the worms as small as possible in diameter, and 
giving the threads in all cases a pitch angle of 20 degrees. The form of tooth 
used was the epicycloidal, while the materials used were hard cast iron for the 
gear and case-hardened open-hearth steel for the worms. 

These Hewes & Phillips worms are plotted in Chart 11 as crosses i, 2, 3, 4, 
of which I is the 15° 15', 2 the 5° 15', 3 the 10° 15', and 4 the 20°, the first 
and last being successes, and the second and third failures. 

In plotting these worms, and all others having pitch-Hne velocities above 
200 feet, the crosses are placed near and above the 200 feet curve. It is 
unfortunate that we have no curves for higher speeds, but Mr. Lewis recom- 
mends the use of the 200 feet line for all higher speeds. Leaders connecting 
different crosses indicate the same worm at different speeds in all cases. The 
letters s and / on the diagram mean success or failure in all cases. 

Fig. 129 is a drawing of a worm 3 (failure) and Fig. 130 shows worm 4 
(success), and no more instructive pair of drawings could be imagined than 
these. The pitches are not far different, and what difference there is is in 
favor of the larger worm. The duty is the same, the gears are of about the 
same diameter, and the revolutions per minute are nearly the same. The 
essential change is in the increase of the pitch angle by a reduction of 
the diameter, and this changed failure to success. 



176 



AMERICAN MACHINIST GEAR BOOK 



The Newton Machine Tool Works use worm gearing in many of their 
machines, notably their cold saw cutting-off machines. In the earlier 
machines of this class the worm had a pitch diameter of 2^^ inches, with a 
pitch of I inch, single thread, the revolutions per minute being 765. These 
figures give a pitch angle of 6° 20', and a pitch-line velocity of 572 feet. This 
machine could be operated, but not with satisfaction on account of the heat- 
ing and short life of the worm. The worm was then increased in lead by 
making it double-threaded, giving a pitch angle of 12° 30', the speed being 
reduced to 500 revolutions per minute, giving a pitch-line velocity of 375 feet. 




PIG. 129. HEWES & PHILLIPS 
UNSUCCESSrUL WORM. 



FIG. T30. HEWES & PHILLIPS 
SUCCESSFUL WORM. 



The change proved to be a great improvement, heavier work than was before 
possible being done after the change without distress or difficulty, and this 
worm has since been applied to a large number of machines with entire suc- 
cess. A still later worm used on these machines has a pitch diameter of 3% 
inches and a lead of 4 inches, triple threads, giving a pitch angle of 18° 15', 
and this is found to be a still further improvement. This last worm is used 
on a wide variety of mchines and at a variety of speeds from 40 to 680 r. p. m,, 
giving pitch-line velocities of from 40 to 685 feet, and with uniformly good 
results. In many cases it is used without an oil cellar, though for compara- 
tively light work. The form of thread used is the involute, and the material 
is hardened steel for the worm and bronze for the wheel. These Newton 
worms appear in Chart 11 as 5, 6, 7, of which 5 is nearly a failure, while 6 
and 7 are entirely successful. The second Newton worm — the one appearing 
in Chart 11 as 6 — is shown in Fig. 131. 

Another habitual user of worms is John Bertram & Sons, of Dundas, 
Ontario, Canada, who employ them in all their planers, and use largely a 
worm of 3.18-inches pitch diameter, 4 inches lead, quadruple threads, the 
speed cutting being 186 and reversing 744 r. p. m. These figures give a 



WORM GEARS 



177 



pitch angle of 22 degrees, and pitch-line velocities of 155 and 620 feet. 
This worm appears in Chart 11 as 8, the vertical position for the higher speed 
being again uncertain. These worms are highly successful, as the writer 
knows from repeated observation. Both worm and wheel are of cast iron, 
the thread being Brown & Sharpe standard. The Bertram worm is shown in 
Fig. 132. In reading this drawing it should be remembered that the con- 
ventional representation of a worm, with the threads shown by straight lines, 
shows a larger apparent pitch angle than the true one, as shown by a true 
projection. 

Another case of failure was a worm drive appHed to a large boring machine, 
the worm being 12-inches pitch diameter, 8-inches lead, quadruple thread, 
speed 80 r. p. m. and above, worm of forged steel, wheel of bronze, oil cellar 




FIG. 131. THE NEWTON WORM AND STEP. 



c:^ 



r*2>6n 




FIG. 132. THE BERTRAM WORM. 



lubrication. These figures give a pitch angle of 12 degrees and a pitch-line 
velocity of 250 feet. This worm is located on Chart 11 as 9. 

Still other cases of change from failure to success are supplied by Mr. Jas. 
Christie, of the Pencoyd Iron Works. The first of these relates to a boring 
machine, which was, by the makers, supplied with a worm drive having a 
worm of 5J'^-inches pitch diameter, ij^^-inches pitch, single thread, steel worm 
and cast-iron wheel, a^verage speed 150 r. p. m. These figures give a pitch 
angle of 5 degrees and a pitch-line velocity of 215 feet. This was a failure, 
but was successfully replaced by a worm of 4%-inches pitch diameter, 2^^ 
inches lead, and the same number of revolutions, which figures give a pitch 
angle of 9° 15' and a pitch-line speed of 190 feet. These two worms appear 
as 10 and 11. This successful worm lies in the region of unsuccessful ones, 
but the influence of the increased lead angle is unmistakable. The fact of its 
success is probably due to the pressure on the teeth being well below the 
working strength, or to the speed being moderate, or both. 

The second case, of which the data were supplied by Mr. Christie, relates 

to two heavy milling machines, in which the cutter spindles were driven by 

worms 6-inches pitch diameter by i3-^-inches pitch, single thread. It was 

found that the cutters could be run much faster than was originally contem- 
12 



178 AMERICAN MACHINIST GEAR BOOK 

plated, and the worms were consequently speeded up to about 500 r. p. m. 
In these machines cast-iron worm-wheels were speedily destroyed, while 
hardened steel worms and bronze wheels would last about a year. Later 
two more machines were built having steel worms and bronze wheels, the 
worms being 43^^-inches pitch diameter by 5-inches lead, quadruple threads, 
speed 280 r. p. m. These worms have been in use six years, and are described 
as being "as good as new." The data given for the first worm give a pitch 
of 4° 30' and a pitch-line velocity of 785 feet. It appears in Chart 11 as 12. 
The pitch angle of the second worm is 19° 30', and its pitch-line velocity is 
328 feet. It appears in Chart 11 as 13. 

Mr. Christie has made many successful changes, of which these are typical, 
and he now uses worms with great freedom and success. His general conclu- 
sion is that good worms begin with those having the pitch about equal to 
the diameter, giving a pitch angle of 17° 15'. 

Another equally striking case of success accompanying an increase of the 
pitch angle is supplied by Mr. W. P. Hunt, of Moline, 111., who says: 

''In building a special double-spindle lathe I wished to use a worm drive, 
and having a single thread ^-inch pitch hob, 2%-inch outside diameter, I 
decided to work to that, and made my gear with 26 teeth, giving a speed 
reduction of 26 to i. The worm was to run at 460 revolutions per minute, 
but upon starting the machine I found it impossible to keep the worm and 
gear cool, and the belts would not pull the cut. 

''Accordingly I decided to make a new worm and hob having the same 
outside diameter as the one first tried, but with double thread and i-inch 
pitch, 2-inch lead and a new gear having 48 teeth, giving me a speed reduction 
of 24 to I , or less than at first. 

'' Upon starting the machine with the new worm and gear, not only did it 
run perfectly cool, but the belts have ample power. We use graphite and oil 
on the worm, and it is not enclosed.' ' 

Mr. Hunt does not give the pitch diameter of his worms, but assuming the 
threads to have been in accordance with the Acme standard, the pitch diam- 
eters are 2% and 23.^ inches respectively, the thread angles being 5° 44' and 
15° 48', and the pitch-line speeds 286 and 271 feet per minute. Mr. Hunt's 
worms are plotted in Chart 11 as 17 and 18. 

Three other cases of successful worms under heavy duty are found in 
milling machines which have been repeated many times. The first two 
worms would ordinarily be described as spiral gears. The shafts are at 
right angles. 

The first of these, which appears as 14 in the diagram, has a pitch diameter 
of 23^^ inches, a pitch angle of 45°, and a speed varying between 180 and 945 
r.p.m., giving pitch-line velocities of 106 to 555 feet per minute. Both gears 
are of cast iron. The second, 15 in the diagram, is of the same style, and has 
the same pitch diameter, with speeds varying between 90 and 472 R.P.M., 
giving pitch-line velocities of 53 to 277 feet per minute. The third, 16 in the 



WORM GEARS 



179 



diagram, is a true worm, 2>^-inches pitch diameter, lead 1.333, triple thread, 
speed 200 to 1,442 r. p. m., bronze wheel and hardened steel worm. These 
figures give a pitch angle of 10° 45', and a pitch line velocity of 118 to 845 
feet per minute. While this worm is entirely successful, it was at first a 
failure. 

LIMITING SPEEDS AND PRESSURES 

A very important point connected with worm design, and one on which 
data are very scarce, is the hmiting pressures for various speeds at which 
cutting begins. The paper by Mr. Lewis contains some information on this 
subject, and the accompanying table supplied by Mr. Christie, from experi- 
ments made by him, supplies the most definite additional data on the sub- 
ject known to the writer. In all cases the worms were of hardened steel 
and the worm-wheels of cast iron. Lubrication by an oil bath. 



Revolutions per minute .... 

Velocity at pitch line in feet 
per minute 



Limiting pressure in pounds 



SINGLE-THREAD WORM 

\" PITCH 

2^^ PITCH DIAMETER 



128 


201 


272 


96 


150 


205 


,700 


1,300 


1,100 



425 
320 

700 



DOUBLE-THREAD 

WORM 2" LEAD 

2^^ PITCH 

DIAMETER 



128 


201 


96 


150 


1,100 


1,100 



272 

20s 

1,100 



DOUBLE-THREAD 

WORM 2)^" LEAD 

43^ PITCH 

DIAMETER 



201 


272 


235 


. 319 


1,100 


700 



425 
498 

400 



Limiting Speeds and Pressures of Worm Gearing. 

There is real need of a comprehensive series of experiments on this subject. 
It is obvious enough that a worm, otherwise well designed, might fail from 
having too high a speed for its load. Were such data at hand it would seem 
that with existing knowledge of the influence of the angle of the thread, worm 
design might be made a matter of comparative certainty. Especially should 
the behavior of worms at speeds above 200 feet per minute be subjected to 
further experiment, as it is frequently necessary to use speeds above that 
figure, and there can be no doubt that higher speeds are entirely feasible if 
suitable pressures accompany them. The speed as a factor should be kept 
in mind equally with the pitch angle. A worm may fail because of too high a 
pitch-line velocity as well as because of too low a pitch angle. 

The number of cases cited is too few for certainty in drawing general 
conclusions, but the testimony is unmistakable in its confirmation of the 
theory of the influence of the angle of the thread. It will be seen that every 
case having an angle above 12° 30' was successful, and every case below 9° 
unsuccessful, the overlapping of the successful and unsuccessful worms in 
the intervening region being what is to be expected in the border region 
between good and bad practice. This band of uncertain results is in fact 
narrower than we would have any right to expect from a collection of data 



l8o AMERICAN MACHINIST GEAR BOOK 

from miscellaneous sources, and could the inquiry be widened in scope the 
width of this band would doubtless be increased. As throwing light on these 
cases, it should be remembered that case i6 is known to have been made 
successful only by careful attention to the material used, the first worms 
made having been failures, and that 3, which is near 16 and was a failure, 
had an excessive speed, while 11, at a lower angle and a success, had a very 
moderate speed. At a higher speed 11 would probably have failed, and at 
a lower speed 3 would probably have been a success. It is believed that 
Chart II points out clearly the nature of the worm problem and the condi- 
tions of success in its solution. 

EFFICIENCY AND TEMPEEATURE 

The unavoidable sliding action between the threads of the worm and the 
teeth of the gear represents lost energy which must be measured by the 
amount of friction heat generated. The elevation in the temperature of the 
lubricating oil about a worm drive should then approximate the efi&ciency of 
the construction. 

The relationship between the efficiency of a journal bearing and its rise 
in temperature is similar and is generally recognized as a reliable indication 
of the efficiency of the bearing. Unfortunately, a number of experiments 
that have been conducted with a view of discovering such relationship in 
connection with worm drives have been rather misleading. This has been 
partly due to inadequate lubrication, but more probably to the fact that both 
liquid and solid friction have been present. 

Comprehensive experiments to ascertain the relationship between effi- 
ciency and temperature of worm drives at the engineering laboratory of the 
Royal Technical High School, Stuttgart, Germany, however, have conclu- 
sively shown that some relationship does exist. The conclusions arrived at 
by the experimenters may be summarized as follows: 

1. The difference in oil temperature is approximately proportionate to 
the tooth pressure when the speed of the worm remains constant. 

2. The tooth pressure decreases according to a definite law with any 
increase in worm speed when the temperature remains constant. (A curve 
depicting such decrease in tooth pressure is of distinctly hyperbolic character.) 

As confirmation of the above deductions, the conclusions arrived at by the 
firm of Henry Wallwork & Co., Ltd., Manchester, England, which has con- 
ducted many individual tests to ascertain the laws governing worm-gear 
efficiency, are of particular interest. A. V. Wallwork summarized these 
conclusions, in a letter that appeared in American Machinist, Dec. 5, 191 2, 
as follows: 

''That the efficiency of a correctly designed gear increases under light 
load and decreases under heavy load, and that the temperature rise of the 
oil gives an exact measure of the power loss in the gear." 



WORM GEARS iSl 

EFFICIENCY OF LANCHESTER WORM GEAR 

The Lanchester worm, which is very similar to the Hindley worm, being 
of the hour-glass variety, has been subject to exceedingly exhaustive experi- 
ments at the National Physical Laboratory of Teddington, England, which 
were briefly described in American Machinist, June 12, 1913. 

The engineers making these tests arrived at the conclusions that the 
efficiency of the worm gear itself depends entirely on the condition of the oil 
film between the worm and the wheel and that the efficiency would remain 
practically constant as long as this oil film is perfect. The fall in efficiency 
of the gear at slow speed, which was quite apparent in all tests, was believed 
to be due in part to the reduction in the quantity of oil carried around by the 
worm. 

In conclusion, the report submitted to the Daimler Co. of England, for 
which the tests were made, says: 

"The conclusion to be drawn from these tests is that the efficiency of the 
gear lies between 93 and 97 per cent, under all circumstances, and, taking 
the normal running speed of the worm at i, 000 r. p. m., the efficiency of the 
gears lies between 95 and 97 per cent., only falling slightly below the lower 
figure when the temperature approaches 100 degrees Centigrade." 

AUTOMOBILE WORM DRIVES 

The prejudice that appears to exist in this country against the worm drive 
for automobiles is not so pronounced in Europe. There does not seem to be 
any reasonable ground upon which to base the objections made by many 
manufacturers in regard to worm-gear transmissions. English manufacturers 
in particular have secured very gratifying success with worm drives. David 
Brown & Sons, Ltd., Lockwood, Huddersfield, England, claim excellent 
results for their worm-gear auto drives — stating that 95 per cent, efficiency 
is easily realized. E. G. Wrigley & Co., Ltd., Birmingham, England, claim 
for their worm drives an efficiency as high as that of a bevel-gear drive at 
normal speed, equal or greater efficient life and practical silence in operation. 

Ralph H. Rosenberg presented a paper before the Society of Automobile 
Engineers, excerpts of which were printed in American Machinist, Feb. 29, 
1902, in which he summarized his conclusions on the worm drive for heavy 
power vehicles in part as follows: 

"My contention is — and I have proved it empirically — that, first, the 
tooth angle and angle of lead or advance must coincide, and second, that the 
advance angle fixes the diameter of the worm. It is determined by extend- 
ing the lines describing the flanks of the teeth to points where the intervening 
distance is equal to the lineal pitch times the number of leads; the distance 
from these points to the pitch circle of the gear is the diameter of the worm. 

"I have adopted the following method for determining the width of the 
gear and face of the teeth. They are described by diverging lines from the 



l82 AMERICAN MACHINIST GEAR BOOK 

center of the worm, including an angle of 120 degrees. Gears made accord- 
ing to this formula will permit a reasonable amount of variation between 
the pitch circle of the worm and the pitch circle of the gear, the surfaces 
remaining complementary. This is not permissible with any other form and 
allows a certain latitude in manufacture. It is absolutely necessary, however, 
to maintain proper relations of the axes. 

COST 

"I have heard it asserted by those conceding the utility and desirability 
of the worm gear that it was an expensive device. My endeavors to ascertain 
upon what ground this assumption was made and what particular item en- 
tered into the consideration of cost were usually met by general statements. 
So I conclude that the cost-of-manufacture information while not as vague 
as that relative to designing is, nevertheless, indefinite. I take the liberty 
of quoting from E. E. Whitney's paper of June, 191 1, relative to the cost of 
worm-gear construction, wherein he states the worm-gear drive is not a cheap 
device and that the indicated efficiency and durability results cannot be 
expected unless the gears are properly designed, constructed of the best 
materials and adequately mounted in high-grade anti-friction bearings. 

''I concede this statement to cover the essential facts generally, but on 
the question of cost take issue, believing the worm gear to be the cheapest 
form of final drive. It is admittedly true that proper design is essential to 
the success of any mechanism, but it does not follow that proper design will 
entail any expense over and above improper design, so far as it relates to 
the cost of manufacture. In my experience I have found that the materials 
used in the worm and gear are not more expensive than those employed in 
the bevel-gear drive or the side-chain drive, where double reductions are 
used. Furthermore, a distinction should be made between experimental 
work and work of actual production where the facilities are provided for 
executing large quantities. In substantiation of my statement I give the 
following data, taken from records covering the cost of producing a worm 
and gear for a 5-ton truck: 

Bronze ring gear blank Si 8.00 

Steel for worm 9.00 

Time to machine worm 2 hours 

Labor on worm, rough- turning 2 hours 

Labor on worm, molding 2 hours 

Labor on worm, grinding 3 hours 

Ring gear, turning 3 hours 

Ring gear, cutting i hour and 

10 minutes 

''The parts are then in condition for assembling. Concerning the desir- 
ability of the worm gear, I am in accord with Mr. Whitney, and can say that 



WORM GEARS 



183 



I have inspected gears after they have run 120,000 miles and found them in 
excellent condition. Granting that the expense of production is higher, it 
is offset by the greater life of the gear." 

There is no question that properly proportioned and constructed worm- 
gear drives can be made with high guaranteed efficiency; that they are com- 
pact and may be constructed for considerably higher ratios without requiring 
undue space for their accommodation; that their wearing qualities can be as 
great as those of a bevel-gear drive; and that they possess other advantages 
of silence, smooth operation, etc. These numerous advantages, which are 
now lessened by no serious drawbacks, should appeal strongly to the automo- 
bile manufacturer, particularly in the construction of auto trucks requiring 
large reduction in speed between the motor and the driving wheels. 

THE HINDLEY WORM GEAR 

An interesting modification of the standard worm gear is frequently 
encountered in elevator service, and is known as the Hindley worm gear or, 
from its form, the globoid gear. The worm differs from the ordinary straight 
worm because, instead of being cylindrical in shape, it is formed somewhat 




FIG. 144. THE HINDLEY WORM GEAR. 

like an hour-glass or spool. The pitch diameter of the worm varies so as to 
coincide with that of the gear for the full length of the worm. The thread 
of the worm is in mesh with several gear teeth at the same time, the worm 
enveloping a section of the gear wheel. See Fig. 144. There is no bottom 
clearance in this gear and the length of tooth and depth of thread are some- 
what greater than in the common gear and worm, thus increasing contact 
surface and therefore decreasing wear. 

The smallest diameter threads, at the center of the worm, evidently 
engage the gear teeth as do the threads in an ordinary worm gear, the con- 



l84 AMERICAN MACHINIST GEAR BOOK 

tact of which is on the pitch line. The threads toward the ends of the worm 
are of greater diameter, however, and as they do not revolve on axes tangent 
to the pitch diameter of the gear, but about the axis of the worm, contact 
is not simply on the pitch line, but on the entire surface of the engaging gear 
teeth. That is, the center teeth are in line contact, while the end teeth are 
subject to surface contact. 

An article by John L. Wood, which appeared in American Machinist, 
June 23, 1 91 4, takes up in detail the method of calculating the strength of 
Hindley worm gears and the methods of calculating them. Excerpts of 
this discussion follow: 

METHOD or CALCULATING STRENGTH 

"The following example is given to show a method of calculating the 
strength of this construction: 

'' In a carriage the theoretical load to be supported by m.eans of two worms 
in the direction of their axes was 25,000 pounds. 

"For the construction the following dimensions for the worm and rack 
were used: 

WORM DIMENSIONS 

Smallest diameter at root of teeth 1.072 inches. 

Corresponding pitch diameter 1-348 inches. 

Corresponding diameter at top of thread 1-587 inches. 

Obhquity of teeth 15 degrees. 

Number of teeth in contact with rack 7 

Pitch = 0.375 inch. 
Lead = 0.75 inch. 

RACK DIMENSIONS 

Thickness = length of tooth = face of tooth 0.95 inch. 

Pitch diameter 12.652 inches. 

Number of teeth in complete circle 106 

Load for each worm and rack 12,500 pounds. 

"Assuming that all of the seven teeth are effective and that each takes 
its proportionate load, we have, 

Load for each tooth = W = 1,785 pounds. 

"Using the Lewis formula for the rack and assuming, as stated before, 

that / = face of tooth = thickness of the rack at the root of the rack teeth, 

we get, 

W = Spfy, 

or 

W I 78t; 
S= — r- = ,, \^ ^ = 42,450 pounds per square inch. 

Spfy 0.375 X 0.95 X0.118 -^ '^^ ^ ^ ^ 



WORM GEARS 185 

''The rack was made of steel with an elastic limit of approximately 53,000 
pounds per square inch, and no trouble was experienced with this rack and 
no change was ever necessary. 

''The worm was made of bronze with a tensile strength of 52,000 pounds 
per square inch. Since its smallest diameter was larger than the face of 
the rack, it might have been supposed that the teeth would have been 
sufficiently strong. This proved not to be the case, and the material was 
changed to steel with an elastic limit of 53,000 pounds per square inch, and 
no more difficulties were experienced. 

"In many other places where this Hindley worm is used it has been found 
that the rack can be calculated by using the Lewis formula and that a safe 
rule for calculating the strength of the worm teeth is: 

"Divide the total load to be supported by the number of teeth in contact. 
Assume this load to be applied at the top of the tooth. Consider the tooth 
a cantilever, of which the base is a line drawn tangent to the root of the 
tooth of the smallest diameter of the worm, the length of this base being 
the distance between the points of intersection of this line and the pitch 
line of the tooth, and the width being the thickness of the tooth at the root. 

" Since this method of calculating the worm is not convenient, the follow- 
ing rules which have shown satisfactory results are given: 

"Instead of considering as the base of the tooth the chord of the pitch 
diameter, which touches the root of the tooth, assume this base to be twice 
the chord subtended by half of this angle. 

" Call this distance/. Then/ = 2 \/Dm, in which D is the pitch diameter 
of the worm, and m is the depth of space below pitch line. Let p = circular 
pitch of worm and rack, then m = 0.3683/?. And 

/ = 2VD X 0.3683/?= 1.2\/JD. 

"Now substitute for/ in Lewis formula, and we have for the worm W = 
i.2Spyy/pD, and for the rack W = Spfy, in which S, p, y have values the 
same as for any other gears, while/ is the width of the rack, and D the pitch 
diameter of the worm; y is the same for the worm, as for the rack. 

" It is evident that a gear of this kind must be made most accurate. This 
is especially so since the teeth in the rack are cut with a hob of as many teeth 
as there are in the worm itself. If any inequalities are found in the shape of 
the teeth in the hob, it will be seen that the rack teeth being cut to suit the 
largest section of the hob tooth and the worm being exactly like the hob, all 
the load might be taken on one tooth only. 

"The following method is the manner of manufacturing these gears at the 
Rock Island Arsenal: 

OPERATIONS FOR MAKING WORM 

"First: Cut off stock J^ inch longer that the dimension required for the 
length of the worm and then center the part. Second: Between centers of 



i86 



AMERICAN MACHINIST GEAR BOOK 



engine lathe rough-turn all diameters 3^^ inch large and all length dimensions 
3^6 inch long. Note: There should be i inch of stock left on the length of 
threaded part to be cut off after thread has been cut. This is to insure against 
any error created by the spring of thread tool on entering and leaving the 
cut and to make sure that the thread tool is cutting on both sides of the 
thread when at the proper length of worm. This rule is important and if 
not followed the worm will have an error in the lead both at beginning and 
ending of thread. Third: Between the centers of the lathe, with special 
fixtures fastened to the carriage, finish the radius. Use the micrometer with 




FIG. 145. SPECIAL FIXTURE CUTTING WORMS OR HOBS. 

double ball points to measure the diameter at the center of the radii or 
smallest diameter. Fourth: Strike a fine line around the piece at the smallest 
diameter. Note: Care must be taken to get this line accurately located, as 
all horizontal measurements are to be taken from this line. A special pointed 
tool is used for this operation. Fifth: Rough out the thread with special 
roughing tool. Sixth: Finish thread with the special thread tool. Seventh: 
Face off the ends of the threaded parts to a proper distance from the center 
line and finish-turn both bearings complete. Fig. 145 shows the lathe set up 
with special fixture-cutting worms or hobs. 



MAKING THE HOB FOR CUTTING HINDLEY WORM-WHEELS 

"First Operation: Between centers of engine lathe rough out the blank of 
the hob, leaving the stock at each end of the part to be threaded equal to 
not less than one-half the pitch. This is important as the end teeth of the 
hob must be equidistant from the center line and have full cutting surfaces. 



WORM GEARS 187 

Second: Between centers in the lathe, with the special fixture, similar to that 
shown in Figs, i and 2, fastened on the carriage, cut the radius to finish, 
using the double half-point micrometer for measuring the diameter at the 
smallest diameter. Third: Strike a line around the piece at the smallest 
diameter. Note: Care must be taken to get this line accurately located as 
all horizontal measurements are to be taken from it. Also, it is used in 
cutting the worm-wheel in proper relation to the hob when a located tooth 
is required. Fourth: Rough-out thread with special tool. Fifth: Finish 
thread with special thread tool. Note : The hob should be larger in diameter 
than the worm by one-tenth of the thickness of the tooth at the pitch line. 
Use the gage or template for depth and width of thread, also, micrometer 
with ball point, sleeve. Sixth: Strike a line parallel with the axis, crossing 
the line described in operation No. 3, in the exact center of the top of the 
thread. This line is essential when a located tooth in the worm-wheel is 
required. Seventh: Mill the flutes deep enough to establish a cutting edge 
at the bottom of the thread, as the hob must finish the face of the worm- 
wheel, also, care must be taken to have the cross lines come in the center of 
the tooth. Mill the flutes square with the helical angle of the thread. 
Eighth : Cut the hob to same length as the worm and remove any teeth on 
each end of hob which would be liable to break off while the hob is cutting. 
Care must be taken to have the same number of teeth on each end of the hob, 
counting from the one with the center fine or cross-line. Ninth: Face the 
ends of the threaded part, measuring from the center line to get the faces an 
equal distance from the center of the radii. Tenth: Back off teeth. Note: 
Care must be taken in this operation to get an equal amount of clearance on 
each side of the tooth. Eleventh: Turn the bearing surfaces at each end of 
the threaded part. Allow about 0.015 inch for grinding after hardening the 
hob. Twelfth: Harden the hob. Note: Temper this hob only at the point 
of the teeth. Care must be taken that the bottom of the teeth is hardened 
as this part of the hob must form the diameter of the wheel. Thirteenth: 
After tempering, set the hob in the centers of the lathe and get the teeth 
running true. Then recenter each end and turn to finish. 

MAKING THE WORM-WHEEL 

"First Operation: Turn up the worm-wheel, leaving the outside diameter 
about 0.02 inch large, to be finished with the hob. Second: Mill the teeth in 
the wheel. Note: The exact centers of the hob must be in the center plane 
of the worm-wheel when teeth are required to be in a fixed relation to some 
other part of the wheel or segment. Turn the hob around in the machine 
until the center of the tooth which has the center line is exactly on top, and 
then set the work in the machine, locating from the center of the cross-lines. 
To get the proper depth of the tooth, measure with a double ball-point mi- 
crometer from the center of the bore to the center of the radii on top of the 
tooth, If the hob is accurately made and located, this measurement will 



i88 



AMERICAN MACHINIST GEAR BOOK 



be found reliable. Note: The worm-wheel must be driven at the proper 
lead by an independent set of gears. 

''The illustration, Fig. 146, shows a machine in operation with the gear 





""'///////'//////////////////'///"/'//ff////f////'//////////(i 




FIG. 147. RACK, WORM AND HOB OF HINDLEY GEAR 

segment being cut with a hob to the Hindley type of tooth. Fig. 147 shows 
a rack, worm and hob of the Hindley type of tooth. 

DIAMETRAL PITCH WORMS 

"If the proper change gears are provided, it is as easy to cut diametral 
pitch worm teeth as any. The proper gears can always be easily calculated 



WORM GEARS 189 

by the rule that the screw gear is to the stud gear as 22 times the pitch of the 
lead screw of the lathe is to seven times the diametral pitch of the worm to be 
cut. For example, it is required to cut a worm of 12 diametral pitch, on a 
lathe having a leading screw cut six to the inch. We have: 

Screw gear _ 22 X 6 _ 11 
Stud gear 7X12 7 ' 

and any change gears in the proportion of 1 1 and 7 will answer the purpose 
with an error of MojOOO of an inch to the thread of the worm. If 22 and 7 
give inconvenient numbers of teeth, the numbers 69 and 22 can be used with 
sufficient accuracy, and 47 and 15, or even 25 and 8, may do in some cases."* 

Care should be taken when using these calculations that the same change 
gears are used to chase both the hob and the worm, as a slight difference in 
the lead of one tooth may prove a serious matter in a worm engaging a large 
gear where several teeth will be in contact. This same precaution applies 
to worms and hobs of a fractional circular pitch. 

It should also be remembered that 4 pitch does not mean 4 threads per 
inch measured on the axis of worm, but 4 threads per inch of diameter of 
the engaging worm gear. The corresponding circular pitch is 0.7854 inch, 
not 0.25 inch. 

* George B. Grant's Treatise on Gearing, Section 120. 



SECTION VII 
Helical and Herringbone Gears* 

The exacting demands of smoothness and silence in operation, long life 
and high efficiency for high-speed gear transmission, such as those imposed 
upon the reduction gears for steam turbines, are simply met by helical or 
herringbone gears only. Such gears are superior to the common spur gear 
in all the requirements made by this trying service. The obliquity of the 
teeth keeps one set in mesh until the following set of teeth is well engaged 
so that at no time is there a sudden transference of load from one tooth to 
the next, as occurs in ordinary spur gearing. The load is gradually put on 
a tooth and as gradually taken off so that the strain on the teeth is kept 
practically constant and the sudden shock of impact, common to spur gearing, 
is avoided. 

In the ordinary spur gear the teeth come in contact over their entire 
length at one time and the whole load is first thrown on the end of the tooth, 
producing the maximum leverage strain as soon as contact takes place. 
This leverage strain is subsequently reduced, but it takes place suddenly 
and, therefore, is much more serious than if led up to gradually. In gears 
with oblique teeth, the load is put on each tooth gradually and as gradually 
removed so that no severe leverage strain is created at any time. 

Helical gears, examples of which are diagrammatically depicted in Figs. 
148-153, possess the drawback of exerting a more or less serious axial or 
side thrust on account of the action of the teeth. This thrust varies with 
the angle of the spiral, so that the highest efficiency is obtained when the 
angle is only great enough to assure the accurate and gradual meshing of a 
set of teeth during the equally gradual releasing of the preceding set of teeth. 
The angle of the spiral need be such only that the end of one tooth will just 
overlap the end of the adjoining tooth. It follows, therefore, that the wider 
the face of a helical gear, the less the angle of spiral need be and the less the 
side thrust produced. 

When employing helical gears it is always desirable to use two gears of 
opposite pitch or spiral in the same shaft, so that the axial thrust of the two 
gears will balance. Ordinarily this is accomplished by the use of duplicate 
sets of gears similarly mounted on common shafts. Where space is limited, 
gears of opposite spiral may be mounted side by side to form virtually one 
gear, such gear being actually a herringbone gear. The exact balancing of 
the helical gear thrust may not always be feasible, nor may it be advisable to 
employ herringbone gears, and in such instances partial balance of thrust 

* See Section XVI for rolled Helical and Herringbone Gears, 

190 



HELICAL AND HERRINGBONE GEARS 



191 



may often be realized. Fig. 150 illustrates such a case; the thrust on shaft 
B is balanced, and that on shafts A and C is reduced to a minimum. 

Two circular pitches are employed for helical gearing: The "normal 
circular pitch," which is the shortest distance between the center of con- 




Fig. 148 



h-.^ 



^ Angle of Spiral 



Fig. 149 



Fig. 150 



Ctrcnlar Pitch 



-\-. 




Fig. 151 




^\ 



Fig. 152 
Normal Circnlar Pitch 




II h-4 

^-^ Circular Pitch 
Fig. 153 

THE DESIGN OF HELICAL GEARS. 

secutive teeth and is measured on an imaginary pitch cylinder, and the 
''circular pitch," which is the distance between the center of two teeth 
in the plane of the gear and which is measured on the pitch circle as for spur 
gears. See Figs. 151, 152 and 153. 

The pitch diameters must be in proportion to the number of teeth and 
both the circular and normal pitches must be the same in both gears of a 
pair. The angle of spiral must also be the same but of opposite hand. 

The form of tooth employed for helical and herringbone gears is the 
involute or a close approximation of that form. 



192 AMERICAN MACHINIST GEAR BOOK 



NOTATION FOR HELICAL AND HERRINGBONE GEARS 

D' = pitch diameter. 
D = outside diameter. 
N = number of teeth. 

p = diametral pitch. 
p' = circular pitch. 
p'^ = normal circular pitch. 
E = angle of spiral. 

5 = addendum. 

/ = clearance. 
W = whole depth of tooth. 

/ = thickness of tooth. 

L = center distance 

= speed ratio 



V 



pair of gears. 



FORMULAS FOR HELICAL AND HERRINGBONE GEARS 



3.I416 

B 

+ I 



= 2 I V , ^ j for gear, (i~^) 



V 

/ B 



X I 



= 2 I F _l_ I for pinion. (i~^) 



V 



P 
D = D' -\- 0.6366^'". (3) 

Cos. £ = ^. ■ (4) 

P 

/« = p' COS. E = ^. (5) 

P' = -^» (6) 

cos.E 

(6-a) 

(7) 

(8) 
(9) 

(10) 

L = 3.1416Z)' cot. E. (11) 

N 
The corresponding spur cutter = -, ^3 (12) 




HELICAL AND HERRINGBONE GEARS 193 



DISCUSSION OF FORMULAS 

Only the pitch diameter and the number of teeth are figured from the 
circular pitch. 

The outside diameter and all tooth parts are figured from the normal 
pitch, the relationship existing to the normal pitch being similar to the 
relationships of spur gears to their circular pitches. 

The angle of the spiral, in practice, is usually selected with a view to the 
spur cutter available. The normal pitch divided by the circular pitch 
gives the cosine of the angle of the spiral. This angle should be kept as low 
as possible, ordinarily not exceeding 20 degrees, in order to avoid undue 
axial thrust. When the angle is fixed as well as the pitch of the cutter, the 
diameter necessary to give the proper combination may be found by first 
determining the circular pitch. A change in the angle to accommodate an 
even number of teeth makes no difference in helical gears. 

The angle of the spiral must be accurately adhered to, for a slight 
deviation will interfere with the proper contact of the teeth. 

The normal pitch should conform as nearly as possible with some standard 
pitch, so it is customary to assume the normal pitch and proportion the angle 
of spiral accordingly. If a variation between the pitch of the cutter and the 
normal pitch cannot be avoided, it is better to have the cutter pitch finer 
rather than coarser than the normal pitch. 

The circular pitch, found by dividing the normal pitch by the cosine of 
angle of spiral, must always correspond to an even number of teeth — i.e.j 
the product of the circular pitch and the number of teeth must represent 
the pitch circumference of the gear in question. 

After settling the pitch of the cutter, the number of teeth for which the 
cutter should be made is found by dividing the actual number of teeth in 
the gear by the third power of the cosine of the spiral angle. 

The spiral lead is the distance traveled by the thread in one complete 
revolution of the pitch circle. As the angle of spiral becomes smaller and 
the form of the tooth approaches that of a spur gear, the lead becomes longer 
until, when the spiral angle is zero, it lengthens to infinity. 



EXAMPLES IN THE DESIGN OF HELICAL GEARS 

I. Required: 

Pair of helical gears cut with a 4-pitch cutter, speed ratio 4 to i and 
center distance i2j^^ inches. 

4-pitch cutter = 0.7854 inch circular pitch (Table I), 
= normal pitch for helical gears. 

13 



194 AMERICAN MACHINIST GEAR BOOK 

Solution : 



DIMENSION 


FORMULA 


GEAR 


PINION 


Pitch diameter 


{i-a and b) 

(2) 

Table I 
(6-a) 
Cosine from (4) 
(8) 

(9) 
(10) 

(3) 
(11) 


20 . 000" 
76 

0.7854" 
0.8267" 
18° 11' 
0.250" 

0.539" 

0.392" 

20.500" 

191 . 292" 

No. 2-4p. 
R. H. 


5 ■ 000" 

19 
0.7854" 
0.8267" 
18^ 11' 
0.250" 

0.539" 

0.392" 

5 -500" 

47.823" 

No. s3^-4P. 
L. H. 


Number of teeth 


Normal pitch 


Circular pitch 


Angle 

Addendum 

Whole depth of tooth 


Thickness of tooth 


Outside diameter 


Lead 


Cutter used 


Hand 











2. Required: 

Pair of helical gears to replace two spur gears of 60 and 1 5 teeth respec- 
tively, 3 pitch, i23-^-inch centers. Speed ratio to remain the same and the 
helical gears to be cut with the same pitch cutter but to have fewer teeth in 
order not to change the center distance. 

3-pitch cutter = 1.0472 circular pitch (Table I), 
= normal pitch for helical gears. 

Solution : 



DIMENSION 



FORMULA 



GEAR 



PINION 



Pitch diameter 

Number of teeth 

Normal pitch 

Circular pitch 

Angle 

Addendum 

Whole depth of tooth. 
Thickness of tooth ... 

Outside diameter 

Lead 



Cutter used 
Hand 



{i-a and b) 

Assumed 

Table I 

(6-a) 

Cosine from (4) 

(8) 

(9) 

(10) 

(3) 
(11) 



20 . 000 

56 

I .0472" 

1 . 1220" 

21° 3' 

0.3333" 

0.9075" 

0.5236" 

20.666" 

163.456" 

No. 2-3p. 
R. H. 



5 ■ 000" 

14 
I .0472" 
I . 1220" 

21^3' 

0.3333" 
0.9075" 
0.5236' 
5.666"' 
40.864" 

No. 7-3p. 
L. H. 



When space permits or when helical gears can be arranged in pairs so as 
to overcome the axial thrust that constitutes the chief drawback of these 



HELICAL AND HERRINGBONE GEARS 



195 



highly efficient gears, they are usually to be preferred to herringbone gears. 
They are easier to machine and possess the always desirable feature of 
greater simplicity. 

A well-balanced helical gear drive is shown in Fig. 154, which illustrates 
the arrangement of gears on a heavy rack cutter. Before the installation 
of these gears, ordinary spur gears had been employed with discouraging 
results. The spurs had broken, worn out rapidly and had proved entirely 
inadequate for the service demanded. The helical gears, on the other hand, 
have proved eminently satisfactory. After several years of operation they 
show no appreciable wear, run smoothly and with little noise, and permit the 
rack cutter to be driven at far higher speed than was formerly possible. 

Helical speed-reduction gears have also proved very successful and some 
remarkably high efficiencies are reported for such apparatus. The De 




FIG. 154. HELICAL GEARS FOR CUTTER DRIVE. 

Laval Steam Turbine Co., which has adopted such mechanism in connection 
with its steam turbines, claims efficiencies as high as 99 per cent, and states 
that an efficiency of q8J^ per cent, is a conservative figure. This enviable 
record being only possible through excellent design, the use of proper materials 
and high-grade workmanship, a brief description of this gear will be of 
interest. 

DE LAVAL SPEED REDUCTION GEAR 

Two helical gears with their pinions are used to overcome the axial thrust 
and to permit a somewhat greater spiral angle to be employed than is custom- 
ary. The angle of spiral is such that several teeth are in contact at the same 
time. The form of tooth used is the involute to secure true line contact as 
long as the tooth is in mesh. 

The gears consist of a rigid cast-iron center or spider upon which steel 
bands of a special grade of steel are shrunk. The gear blanks are carefully 
mounted and trued up before the teeth are cut in order to insure accuracy. 

The pinion is cut directly on the pinion shaft, which is a special nickel- 
steel forging that is oil-tempered to the desired degree — the pinion having 



196 



AMERICAN MACHINIST GEAR BOOK 



to be considerably harder than the gear bands to assure uniform wear and 
long life. 

After the gear and pinion have been cut, they are carefully polished to 
remove any tool marks and unevenness. This is accomplished by running 
the gear and pinion with similar and as accurately cut ''dummy" gears. 
This polishing process naturally adds to the cost of manufacture, as the 
*' dummy" gears rapidly deteriorate from wear and have to be discarded. 
The added cost of polishing is well warranted, however, as it greatly increases 
the life and efficiency of the finished gear. 

The bearings, lubrication and details of the gear case for such speed 
reducers are not dissimilar to those of any other high-grade speed-reduction 
gear. 

Special flexible couplings for connecting the pinion shaft to the steam 
turbine and the gear shaft to the machine to be driven are necessary, of 
course, for any mis-alignment would be suicidal to the high efficiency 
demanded. 

Speed reduction gears built with this care have been opened up after more 
than 10 years of constant and exacting service and have shown no wear. 



HERRINGBONE GEARS 



A double-helical or herringbone gear avoids the axial thrust of the single- 
helical gear which is not properly balanced, consisting as it does of virtually 






FIG. 15s. 



FIG. 156. 



FIG. 157. 



THE DESIGN OF HERRINGBONE GEARS. 

two gears exerting axial thrusts in opposite directions and thereby nullifying 
any unbalanced side pressure. 

The formulas for herringbone gears are the same as for helical gears, but 
the angle of spiral can be made considerably more obtuse. In fact, the angle 
of the teeth with the axis of the gear is only limited by constructional diffi- 
culties. Angles of 30 degrees are quite common and angles as great as 45 
degrees are quite frequently used. 



HELICAL AND HERRINGBONE GEARS 



197 



Fig. 155 shows a common type of herringbone gear, and Figs. 156 and 157, 
modifications that are frequently resorted to to faciUtate manufacture. The 
gear shown in Fig. 155 really consists of two helical gears fastened together. 
The gears are cut separately and not connected until after machining. This 
simplifies the cutting of the teeth but also adds to the cost of manufacture. 
The type shown in Fig. 156 differs only in that the rim is made in two pieces 
and separate from the spider. The same ease in cutting the teeth is thus 
secured as in the other type and the added advantage of simplifying 
replacements should the gear teeth be damaged or wear out. 

The grooving of the type shown in Fig. 157 allows the gear to be made 
in one piece, as the cutter can be run out at the groove. In this type of con- 
struction, the teeth may be staggered to somewhat lessen the width of the 
central groove. 

The great difficulty of accurately cutting herringbone gears so that they 
may be interchangeable has led to the adoption of many makeshifts. When 
the gear is made in one piece, the pinion is sometimes made in two pieces to 



Eaw-hide Washer 




FIG. 158. ARRANGEMENT OF PINION. 

facilitate the proper engagement of the teeth. The pinion is run with the 
gear and not key seated until after the teeth have accurately adjusted 
themselves to the position of the teeth in the gear. 

Another plan that was suggested by ''Attic" in the American Machinist 
consists of placing a washer of some elastic material between the two halves 
of the pinion, as shown in Fig. 158, so that they can adjust themselves by a 
slight axial movement, a movement along the shaft having the same result 
as if the half gears were slightly turned about the shaft. 

INTERCHANGEABLE SYSTEM FOR HERRINGBONE GEARS 



Percy C. Day presented an interchangeable standard for herringbone 
gears before the American Society of Mechanical Engineers. 

The proposed standard, which has been adopted by at least two important 
manufacturers of herringbone gears, is as follows: 

Tooth shape Involute. 

Pressure angle 20 degrees. 

Spiral angle 23 degrees. 



1 98 AMERICAN MACHINIST GEAR BOOK 

^. , ,. , , , . number of teeth 

Pitch diameter (20 teeth and over) = -^p , — -. — t— 

diametral pitch 

_,,,,. , . . . number of teeth +1.6 

Blank diameter (20 teeth and over) = -r. , — . — 7 

diametral pitch 

^. , ,. , , , s o.Qc; X number of teeth + i 

Pitch diameter (under 20 teeth) = — -^ r^ , — -. — -, 

diametral pitch 

^, , ,. , , , . o.Qi; X number of teeth + 2.6 

Blank diameter (under 20 teeth) = — ^-^ ^ — -. — 

diametral pitch 

Addendum ' • 

Dedendum ' • 

I 8 
Full tooth depth ' • 

Working tooth depth .^ • 

Standard face width for gears with pinions of not less than 25 teeth, six 
times the circular pitch. 

Face widths for high-ratio gears with small pinions, six to twelve times 
the circular pitch. 

When a pinion of less than 20 teeth is used with a standard gear, the 
center distance must be slightly increased to suit the enlargement of the 
pinion. If it is desired to keep the center distance to the standard dimen- 
sions, the gear diameter may be reduced by the amount of the enlargement 
given to the pinion. 

For example: If a pinion of 10 teeth, 5 diametral pitch (D.P. ) is to mesh 
with a gear of go teeth at lo-inch centers: 

T^- 1 T <• • • (o-9S X 10) + I . , 

Pitch diameter of pinion — -^ = 2.1 inches. 

Enlargement over standard pinion = 0.1 inch. 

Pitch diameter of standard gear = ^^ = 18 inches. 

5 

Reduced pitch diameter of gear = 18 — o.i = 17.9 inches. 

Center distance -^--^ ^ = 10 inches. 

2 

Strictly speaking, there can be no enlargement for reduction of the pitch 
diameter in a pinion or gear of given pitch and number of teeth. It is con- 
venient to assume such enlargement or reduction, however, when using teeth 
of long and short addenda but standard depth. 

MACHINING HERRINGBONE GEARS 

Three general methods of cutting herringbone gear teeth are in vogue: 
(i) A double-hobbing machine of special design is employed by which the 



HELICAL AND HERRINGBONE GEARS 



199 



right- and left-hand teeth are cut simultaneously. In such machines the 
cutting profile of the hob teeth, cutting their way into the gear blank on a 
diagonal line conforming to the obliquity of the teeth, are modified from the 
true involute form in order that the space cut may closely approximate the 
correct involute profile in the direction of rotation. For extreme accuracy, 
therefore, a uniform obliquity of tooth is necessary, or else a special hob for 
each angularity of tooth. (2) The method second in importance is that of 
planing the teeth. Planers are employed which use a cutting tool having 
the same section and form as the finished tooth space, or machines with 
templets for guiding simple planing tools are used. (3) Herringbone gears 
of the smaller sizes are also cut with single rotary cutters. 

HOBBING PROCESS FOR CUTTING HERRINGBONE GEARS 

Machines of two general types are employed in the bobbing process. The 
working principle of the type shown diagrammatically in Fig. i59-<zisas 
follows : 

The gear blank a is mounted on the vertical mandrel c, which is rotated by 
the face plate b through the spindle d actuated by the driving worm gear e. 




Fig. 159-a. DIAGRAM OF HOBBING PROCESS 
FOR CUTTING DOUBLE HELICAL GEARS. 

The hobs// are mounted in vertical slides gg, which move up and down on the 
standards hh. These standards are mounted so as to slide on the bed of the 
machine J and are provided with micrometer screws for adjusting the depth 
of cut. 

The rotating speed of the gear blank is controlled by a train of wheels 
which operate a differential gear so as to give the required spiral lead to the 
teeth. The process is entirely automatic and evades the necessity of 
inclining the hob axes, the lead being governed by the speed at which the 
gear blank is revolved. This allows the hobs to be set at right angles to the 
axis of the gear blank and the same hobs to be used for gears of different obliq- 
uity of tooth, provided extreme accuracy is not essential. 

Fig. 159-& shows the other type of bobbing machine for cutting herring- 
bone gear teeth. This single-headed machine has two hobs, a right- and a 
left-hand one, carried on a single saddle. The hobs rotate in opposite 



200 



AMERICAN MACHINIST GEAR BOOK 



same 



directions and are fed downward, cutting both halves of the gear at the 
time. The cutting pressures are opposed and neutrahzed through the 




riG. 159-^. HOBBING MACHINE FOR CUTTING DOUBLE HELICAL GEARS. 




FIG. I59-C. MACHINE FOR PLANING DOUBLE HELICAL GEARS. 

section of the gear being cut, thus reheving the machine of any such 
unbalanced stresses. 



HELICAL AND HERRINGBONE GEARS 20I 

The lower hob is carried on a spindle, while the upper hob is carried on a 
slide which is vertically adjustable. This allows the hobs to be set for cutting 
gears of various widths, and the adjustable features of the hobs permit them 
to be set so as to cut teeth with apexes on the center line of the gear or teeth 
of the staggered variety. 

The desired depth of cut is fixed by horizontal adjustment of the work 
carriage, after w^hich the cutting operations require little attention, the 
retardation and acceleration of the hobs, the rotation of the work table or 
carriage, etc., being entirely automatic. 

PLANING PROCESSES FOR CUTTING HERRINGBONE GEARS 

Two general schemes of planing herringbone gear teeth are employed: 
First, that in which a tool of the exact section and form of the space to be cut 
is used; and second, the one in which templets are employed to guide the 
simple cutting tools employed — the ''former" method. The first method is 
only employed for cutting comparatively small gears with circular pitches 
usually under i inch, the second for machining larger gears of coarser pitch. 

A large herringbone gear planer in which the tooth profiles are shaped by 
formers is shown in Fig. 1 59-c. The two tool saddles move toward each 
other on the cutting stroke, both halves of the gear being cut at the same 
time by tools advancing from the outer edges of the gear to the center apex 
line of the tooth. The cutting pressures of the tools are thus resisted by the 
gear itself and neutralized, instead of being taken up by some part of the 
machine. The main shaft carries two worm-wheels, the larger for indexing 
and the smaller for rotating the gear blank. Two sets of change gears, one 
for the angle of the tooth and the other for the number of teeth, together 
with a reversing drive constitute the special features of this particular planer. 

MILLING PROCESS FOR CUTTING HERRINGBONE GEARS 

The following description of the milling process for cutting herringbone 
gear teeth is taken from an article in American Machinist by Percy C. Day, 
as is also the test under the three following sub- titles. 

''In the milling process the teeth are sometimes cut by means of end mills 
formed to the tooth shape on the normal section. The working principle of 
the machines usually employed is shown in the diagram. Fig. 160. The end 
mill a is supported by the saddle b, which traverses the bed c. The mill is 
driven by the bevel gears d from the splined shaft e and driving cone /. The 
feed and differential motions are driven from e through speed cones or gears g 
and clutch h. The traverse of the saddle b is actuated by the feed screwy. 
Motion is also transmitted from 7 through change wheels k, reversing gears I, 
dividing change wheels m, worm 0, dividing wheel w, and work spindle p to 
the blank q. 



202 



AMERICAN MACHINIST GEAR BOOK 



"While the end mill traverses from one edge of the blank to its center, the 
blank is rotated through an angle which gives the requisite spiral form to 
the tooth. The saddle then operates a stop which is in connection with the 
reversing gear /, and the rotation of the blank is reversed until the end of the 
cut. A quick-return mechanism, not shown in the diagram, comes into action 
at the end of the cut, and the mill is returned to the starting position. The 
dividing mechanism m is then operated by hand, and the cutting process is 
repeated on another tooth. 



^ I ^ 





FIG. l6o. DIAGRAM OF MILLING PROCESS 
OF CUTTING DOUBLE HELICAL GEARS. 



a := Half Circumferential Pitch fl=a- 

b = Space cleared by Cutter at Ihe Turn 

The shaded Portion must be removed before the 
Teeth will Meeb 

FIG. l6l. DIAGRAM OF SPACE NOT CUT 
BY MILLING PROCESS OF CUTTING 
DOUBLE HELICAL GEARS. 



"The end-milling process can be readily adapted for cutting double helical 
bevel gears. 

"The disadvantages of the process are principally of a practical nature. 
End mills are small tools, and are liable to rapid wear. Since the teeth are 
cut singly, any wear on the mill causes a change of tooth shape and thickness. 
The reversal of the angular motion of the blank while cutting proceeds allows 
the inevitable backlash in the mechanism to take effect in a manner which 
is not conducive to accurate work. The cutter must be formed to the normal 
tooth section, and has not the circumferential shape of the teeth which it cuts. 
The width of the tooth space at the apex corresponds to the normal instead 
of to the circumferential pitch, hence the space must be cleared out by hand 
or in a separate operation (see Fig. i6i). 

"The tendency to wear is greater when the end mills are small, and wheels 
on this system are generally made of coarser pitch than is really necessary or 
even desirable from the user's point of view, in order to minimize the manu- 
facturing difficulties by the use of large mills. 



ADVANTAGES OF THE DOUBLE HELICAL SYSTEM 

"The adoption of the double helical principle in gearing, if properly 
applied, reduces noise to a minimum and practically eliminates vibration 
without any necessity for departure from sound mechanical principles. In 



HELICAL AND HERRINGBONE GEARS 



203 



A-=i 




Pa=l 




t„=l 



tn=0.71 






this type of gear, pinions may be chosen of sufficient hardness to wear evenly 
with the wheels, and soft materials do not enter the proposition. This is due 
to the absolute continuity of engagement which is characteristic of double 
helical gears when accurately cut and correctly designed to suit the working 
conditions. The effect of vibration is not by any means confined to the gears 
themselves, but acts injuriously on the shafts and machinery connected 
therewith. Many failures of haulage and 
other gear-driven shafts have been directly 
traced to this cause. 

"Consider a pair of wheels transmit- 
ting 100 horse-power with an efficiency of 
96 per cent. If we assume only one- 
tenth of the lost energy to be dissipated 
in vibration which is absorbed in the wheel 
shaft, the result is somewhat surprising. 
Under such conditions the shaft is called 
upon to absorb energy at the rate of 
nearly eight million foot pounds during 
each working day of 10 hours' duration. 
The result is finally expressed in crystal- 
lization of the shaft material. . . . 

"Another interesting application of 
machine-cut, double helical gears is the 
reduction of speed from high-power steam 
turbines. No other type of gear can be 
used for this class of work, because abso- 
lute smoothness of action is essential. 
The essence of this problem is to avoid 
excessive velocity by keeping the pinion 
diameter small, but at the same time it 
is undesirable to reduce the number of 
teeth below a certain point because abso- 
lute continuity of engagement must be maintained. The result of these 
conditions is that the gears must be of extremely fine pitch and great 
relative width. For example, a set of gears recently constructed for a 500- 
horse-power steam turbine, to reduce from 3,000 to 300 revolutions per min- 
ute, were 4 diametral pitch with face width 10 inches and pinion of 19 teeth. 

"There is probably no field of application for double helical gears which 
offers such substantial advantages as for driving machine tools. In most 
modern machine shops there is a tendency to dispense with shafting as far as 
possible, and to drive the tools individually from separate motors. This 
method allows a more economical distribution of machines over the available 
floor space, and leaves the space overhead clear for rapid handling. On the 
other hand, motor-driven tools require far more gearing than when the drive 



p^i 






fn=1.10^ 




'^ 




Pq = Circumferential Tooth Pressura 

Pn^ Normal Pressure 

Pa=^ Aiial Pressure 

tc= Circumferential Tooth ThlckneSB 

i n= Normal Thickness 

f = Circumferential Stress 

f n=' Normal Stress 

FIG. 162. COMPARISON OF TOOTH PRES- 
SURE, THICKNESS, AND RELATIVE STRESS 
FOR THE TOOTH ANGLES OF DOUBLE 
HELICAL GEARS. 



204 



AMERICAN MACHINIST GEAR BOOK 



is effected by belts, and it has been found difficult to obtain uniformity of 
motion under the new conditions. If machine-cut, double helical gears are 
used for this purpose, the quality of the work turned out is much improved 
and, by reason of reduced vibration, higher speeds and coarser feeds can be 
employed. 

"The diagram. Fig. 162, shows the relative normal tooth pressures, pitch 
line sections and stresses for angles of 23, 45, and 60 degrees. 

''One of the greatest advantages of machine-cut, double helical wheels is 
to be found in their adaptability for high ratios of reduction. The number of 
teeth which can be used with success in the smallest pinion lies far below 
the practical limit for straight spurs, and pinions of four or five teeth are 
by no means uncommon for special purposes. Since, however, the pitch 
can be made very fine, it is rarely necessary to reduce the number of teeth 

so far, and most high-ratio gears are made 
with pinions of 11 to 20 teeth. Pinions for 
high ratios are generally cut solid on their 
shafts, in order that the diameters may be 
kept low to bring the wheels within reason- 
able proportions. 

" Single wheels and pinions will transmit 
heavy powers with ratios between 10 and 
20 to I, so that they can be used in place 
of worm gears or double trains of ordinary 
spurs. As against worm gears the gain lies 
in the direction of increased efficiency and 
life. A set of double helical gears with 20 to 
I ratio has an efficiency of about 95 per cent, 
against a maximum of about 80 per cent, 
for a worm gear of equal ratio. 



AN INTERESTING APPLICATION OF 
DOUBLE HELICAL GEARING 



"An interesting example of this differ- 
ence came under my notice a short time ago. 
A worm gear of first-class manufacture and 
modern design had been in use for some 
2y<i years, driving a deep-well pump from 
a 50-horse-power motor with reduction 480 
to 22 revolutions per minute. This gear was 
replaced by a double train of machine-cut, 
double hehcal wheels, the ratios being 480 to 60, and 60 to 22. The records 
of power consumption and pump duty were regularly kept, and after the new 
gear had been running for a year the figures showed a net saving of over 17 
per cent, in its favor as against the average for the whole life of the worm 




FIG. 163. TYPICAL HIGH-RATIO 
GEARS WITH STAGGERED TEETH. 



HELICAL AND HERRINGBONE GEARS 



2oq 



gear. It was also shown that the efficiency of the double helical gear had 
actually improved after a year's daily work, while the worm gear had 
steadily deteriorated in this respect from the day it was started. 

''Fig. 163 shows a set of double hehcal gears that are representatives of 
their design and construction. 



IMPORTANT POINTS IN APPLYING DOUBLE 
HELICAL GEARS 

"In conclusion it is desirable to add a word of caution to those who are 
about to adopt this class of gear for the first time. It must not be forgotten 
that there are three fundamental points of difference between machine-cut, 
double helical wheels and ordinary spur gearing: 

" (a) The pitch is finer. 

" (b) The face width is greater. 

" (c) The tooth pressures are generally higher. 

"To insure satisfactory working it is necessary that the shafts shall be 
parallel, true, and rigidly supported. The center distance must also be 
adjusted with great care on account of the fine pitch and small clearances 
allowed. Motor pinions of high-ratio gears should be mounted on extended 
shafts with an outer bearing. Anything in the nature of an overhung drive 
should be avoided wherever possible. 

"To avoid undue wear from magnetically controlled end-thrust in motors, 
the pinions should be mounted on two parallel feathers set at 180 degrees 
and carefully bedded to the key ways (see 
Fig. 1 64) . The pinions should be a good 
tight fit on the motor shafts, but there 
should be just sufficient freedom to allow 
them to move along under the influence of 
continued side pressure, so that the motor 
armature can reach a neutral position 
where the pressure ceases. It is unneces- 
sary to allow the pinions to slide freely on the shafts, and if this is done there 
may be trouble from excessive wear of the keys and keybeds." 

DETERMINING LEAD AND ANGLE FROM SAMPLE 

To produce a herringbone gear to operate with a sample, the calculations 
for which are unknown, is generally a matter of cutting and trying until a 
satisfactory gear is produced, as for herringbone or helical gears the angle 
and lead must be exceptionally accurate, the teeth having contact their entire 
length, and a slight error is noticeable. There is more or less leeway for 
spiral gears, but the method as described can be applied to them as well. 

Cover the points of the teeth in sample with an application of lampblack. 





FIG. 



164. AN IMPROVED METHOD OF 
KEYING FOR GEARS. 



2o6 



AMERICAN MACHINIST GEAR BOOK 



or anything that will make a clear impression on a piece of clean white paper. 
Roll the gear thus treated on the surface of the paper, being careful not to 
allow it to slip, until a sharp impression of the points of the teeth is made, as 
illustrated in Fig. 165. This will represent a development of the teeth at 
the outside circumference. 

The angle of the teeth at the outside circumference may then be measured 
with a protractor by extending the lines of the tooth as developed on the 
paper. 




riG. 165. IMPRESSION MADE BY ROLLING 
SAMPLE HERRINGBONE GEAR. 



/ = Face of herringbone gear. 

a. = Length of the tooth from center of face. 
j8i = Angle of spiral at outside diameter. 

j8 = Angle of spiral at pitch diameter. 

L = Lead of spiral. 
C\ = Outside circumference. 

C = Pitch circumference. 



COS. j8i = 
For helical gears this formula would be: 



0.5/. 



COS. /3i = 



/. 



The next step is to find the lead 



L = 



Ci 



tan. B] 
As the lead is necessarily the same at the outside diameter as it is at the pitch 



HELICAL AND HERRINGBONE GEARS 



207 



diameter of a helical or spiral gear when cut with a rotary cutter, the angle 
of spiral at the pitch line may be found by formula 4. 

C 



Tan. jS = 



L 




American Machinist 



FIG. 166. DIAGRAM OF ANGLES OF HERRINGBONE GEAR. 



The fact that the lead is the same at all points when cutting a spiral, helical, 
or herringbone gear cutter, using a single rotary cutter, makes the solution 
of this problem a simple 
matter. Fig. 166 is self- 
explanatory. 

This being the case, it is 
apparent that such a cutter 
cannot reproduce its own 
shape in the gear blank, as 
to do this the angle and lead 
must be proportional to all 
parts of the tooth. When 
the teeth are generated this 
condition is fulfilled and the 

angle at the pitch line will be proportional to the pitch and outside circum- 
ferences, or: 

EFFICIENCY AND STRENGTH OF HERRINGBONE GEARS 

The efficiency of accurately cut herringbone gears for ratios up to 10 to i 
is about 98 per cent. For greater speed ratios, their efficiency shows a 
slight falling off, but if a single reduction is not abnormal and the gear and 
pinion well mated, the efficiency should not be less than 96 per cent. Even 
better records have been realized by gears which have been particularly 
accurately cut, polished and run in an oil bath. Frequently an efficiency of 
99 per cent, or even slightly higher has been obtained by such gears with 
speed ratios not exceeding 10 to i. 

The life of herringbone gears is far greater than that of even the most 
carefully cut spur gears, and though accurate data are hard to find on this 
subject, it may be safely stated that the usual life of a carefully cut herring- 
bone gear is at least three or four times that of a similar spur gear. 

W. C. Bates, Mechanical Engineer, Fawcus Machine Co., prepared for 
American Machinist a comprehensive article on the design and strength of 
herringbone gears, from which the following important points are abstracted. 

Due to the advantages of the herringbone construction, compared to that 
of spur gears, the load is always distributed over more than one tooth, the 
transference of load from tooth to tooth is without shock, the bearing pres- 
sure angularly placed on the tooth diminishes the strain on the root of the 



2o8 AMERICAN MACHINIST GEAR BOOK 

tooth, the tooth is shorter and therefore more sturdy, and the spacing of the 
teeth and tooth form are more accurate on account of the hobbing process 
employed in cutting the teeth, so that the indeterminate tooth stresses are 
at a minimum when running at high speeds. 

Such advantages naturally allow considerable modification of the well- 
known Lewis formula for the strength of gears, as such empirical formula 
pertains to the strength of herringbone gears. 
It then becomes: 

c / 1,200 

1,200 + V 
where 

S = allowable stress per square inch at a speed of " feet per minute. 
s' = static stress = 8,000 pounds per square inch for cast iron, 

= 20,000 pounds per square inch for steel, 
V = speed in feet per minute. 

MODIFIED HERRINGBONE GEARS 

An efficient modification of the standard herringbone gear may be con- 
structed with a form of tooth that relies only upon its rolling action and 
eliminates all sliding contact. A tooth of the regular involute form on the 
pitch line is employed with the surplus metal above and below this contact 
line cut away. Such a gear will run as smoothly as the standard herringbone 
with full tooth section as long as no appreciable wear takes place. This 
naturally limits the practical value of the type, but it is of interest as repre- 
senting the ideal in gearing action. 

A radical departure from any ordinary type of gear has recently been put 
on the market by the R. D. Nuttall Co., Pittsburg, Pa., which has been 
designated as ''The Circular Herringbone." A description of this gear 
appeared in American Machinist, Oct. 16, 1913, from which the following 
excerpts are taken. 

"It has a continuous tooth curved across the gear face, the curve being a 
circular arc. This approximates the shape of a herringbone tooth, hence the 
name "The Circular Herringbone." The tooth profiles at the middle of 
the face are true involutes; other profiles vary slightly from this, but are 
close approximations to the involute form. Though any tooth proportions 
can be cut, those standardized are: A pressure angle of 20 degrees, addendum 
0.25, dedendum 0.25, clearance 0.05, working depth 0.5, and whole depth 
0.55 of circular pitch." 

These gears are generated, the cutter and blank rolling together during 
the process of cutting in such a manner that the line of tangency is along 
the pitch line of the cutter and the pitch surface of the blank. 

HERRINGBONE BEVEL GEARS 

The commercial development of a practical system of rolling gears from 
metal blanks heated to a semi-plastic condition — Anderson process— ^has 



HELICAL AND HERRINGBONE GEARS 209 

made possible for the first time the production of herringbone bevel gears 
possessing all the merits of the efficient herringbone spur gear and in addition 
the capacity of absorbing in themselves the axial thrust common to other 
types of bevel gears. This noteworthy achievement in gear development 
gives promise of greatly extending the field for bevel gearing, particularly as 
herringbone bevels are as readily and cheaply made as ordinary bevel gears — 
see Section XVI. 



14 



SECTION VIII 

Spiral Gears 

Before going into the matter of calculations it may be well to direct the 
readers to a careful consideration of the accompanying perspective sketches 
originally pubHshed in American Machinist, October ii, 1906, by H. B. 




Fig. 169 



Fig. 170 
SPIRAL-GEAR DIAGRAMS. 



Fig. 171 



McCabe. ''In Figs. 167 to 171, inclusive, the driving gear of each pair 
is shown as if transparent, the teeth being represented by lines. Fig. 167 
shows a pair of gears on shafts at right angles and Fig. 168 a pair on parallel 
shafts. Note that in Fig. 167 both spirals are left hand, while in Fig. 168 
one is left and the other right hand; that is, in the first case they are the same 
hand and in the second case they are opposite hands. (The word hand as 
here used has the same significance as in the case of threads.) It is evident 



210 



SPIRAL GEARS 2ii 

that these two are fixed conditions for shafts respectively at 90 degrees and 
parallel. 

''Now when the shafts are at any angle between 90 degrees and o degrees 
either of these conditions may exist; that is, the spirals may be both the same 
hand or they may be opposite hands. This may be made plain by observing 
carefully Figs. 169, 170, and 171, in which the shafts are at an acute angle, 
all conditions in the three views being exactly alike except that the teeth are 
at different spiral angles in each. Note that in Fig. 169 the spiral angle of 
the driver is the same as the angle of the shafts which makes the follower 
a plain spur gear. Also note that the spiral of the driver is left hand. Now 
letting the spiral of the driver remain left hand, but increasing its angle a 
little we have the condition of Fig. 170. By decreasing it a little we have 
the condition in Fig. 171, making in the first case the spirals opposite hands 
and in the second case the spirals the same hand. 

The lines OA and OB in these figures are drawn parallel to the shafts 
and the line OC is drawn tangent to the spiral of the teeth and makes with 
OA and OB respectively the spiral angles of the driver and of the follower. 
Note that in Fig. 171 the angle of the shafts AOB equals the spiral angles 
AOC + ^OC, and in Fig. 170 the same angle AOB equals AOC — BOC 

RELATION OF SHAFT AND SPIRAL ANGLES 

"The following general rules are now evident: 

" I. When the spirals are the same hand the angle of the shafts is the sum 
of the spiral angles. 

''2. When the spirals are opposite hands the angle of the shafts is the 
difference of the spiral angles. 

"3. When the spiral angle of one gear is the same as the angle of the 
shaft the spiral angle of the other will be zero, making it a plain spur gear. 

''4. When the shafts are at right angles the spirals must both be the same 
hand. 

"5. When the shafts are parallel the spirals must be opposite hands. 
(Helical gears.) 

"6. When the shafts are at any acute angle the spirals may be either the 
same hand or opposite hands." 

The following is an extract from an article on spiral gearing originally 
pubHshed in American Machinist by F. A. Halsey: 

*' Spiral gears are not to blame for the undoubted fact that they are some- 
what troublesome to lay out, the difficulties of the problem being due to the 
limitations of workshop facilities and not to the geometrical nature of the 
gears themselves. It is easy to understand and explain the action of an 
existing pair of spiral gears. More than this, it is easy to lay out a pair of 
such gears which shall exactly meet all the conditions of the case except one; 
they cannot, except through rare good luck, be made with the appliances at 



212 



AMERICAN MACHINIST GEAR BOOK 



hand. To be more specific, the circumference cannot usually be divided into 
an exact whole number of teeth by any stock cutter, and the real problem 
becomes the readjusting of the diameters of the gears and the angle of the teeth, 
so that stock cutters shall make an exact whole number of teeth. 

" With spur gears it is only necessary to multiply the (circular) pitch of the 
cutter by the number of teeth to be cut to obtain the circumference of the 
gears. With spiral gears this operation gives the length of a portion of a spiral 
or, more properly, helix, wound upon the pitch surface. We do not know the 
angle of this helix, the diameter of the pitch cylinder upon which it is 
wrapped, or even what part of a complete turn the known portion comprises. 
The length is known for each gear and nothing more, and it becomes a 
matter of trial to find the diameters of the gears and the helix angle to suit 




FIG. 172. 




this portion of the helix and at the same time to fill the required center 
distance. 

^'Fig. 172 is a conventional representation of the pitch surface of a spiral 
gear, the surface being extended beyond the limits of the gear in order that the 
two helixes with which we are concerned may be shown. The first of these, 
abcdef, is the tooth helix and the second, aghdip, is the normal helix. 
The tooth helix is of importance because it defines the angle of the teeth. 
Given the diameter of the pitch surface, the helix may be defined by the angle 
kal or by the length af, in which it makes a complete turn — that is, by its 
pitch. For the determination of the speed ratio of a pair of gears the former 
method is the more convenient, but the tables supplied with universal milling 
machines which are used in setting up the machine employ the latter method. 

"In all spiral gear problems we have two pitches to deal with — the pitch 
of the tooth helix and the pitch of the teeth. The latter may be measured in 
several ways. First is the value an measured on the circumference or the 
circular pitch, which is analogous to the pitch of spur gears; second is the value 
ao measured on the normal helix or the normal pitch, for which the cutters 
must be selected; third is the value ar measured parallel with the axis or the 
axial pitch. Since the cutters must be selected with reference to the normal 
pitch, the length of the normal helix is naturally of importance in conection 
with the number of teeth in the gear. The normal pitch multipHed by the 
number of teeth must naturally equal the length aghd of this helix measured 
between its intersections a and d with the helix of a single tooth. Note that 



SPIRAL GEARS 



213 



the length of the normal helix to be considered is the length aghd between 
its intersections with the tooth, and not the length aghipq of a complete 
turn around the cylinder. That this is true may be seen by reference to Fig. 
173, in which the angle kal is nearly a right angle. It is apparent from this 
illustration that the length of the normal helix from a toe? takes in all the teeth, 
and that ao, multipKed by the number of teeth, must equal ahpd and not 
ahpq. This length ahpd is always less than ahpq, and usually much 
less. Fig. 174, A, is a development of Fig. 173 on a reduced scale, ad being 
the developed length of the normal helix. Fig. 174, B^ and Fig. 174, C, show 





FIG. 175. 

how with the same circumferential pitch and the same number of teeth but a 
reduced value of the angle kal, the length of the normal helix which cuts all 
the teeth grows shorter until it may make but a small part of a complete turn 
around the cylinder. It is clear that in all cases the line ad cuts all the teeth 
precisely as does the circumference aa, which goes completely around the 
cylinder. It is also clear that if the normal pitch is decided upon at the start, 
a diameter of cylinder and a helix angle must be found such that the normal 
pitch, multiplied by the number of teeth, shall equal the length of the normal 
helix between two intersections with the tooth helix. 

"It is natural to ask. Why not employ the circumferential pitch and so 
deal directly with the circumference instead of the normal helix? Because 
we do not know what it is. The normal pitch is determined by the cutter 
used, while the circumferential pitch depends also upon the helix angle, and 
until this angle is known the circumferential pitch is not known. 

"In the extreme case of a spiral gear in which the helix angle is so small 
that the gear becomes a single thread worm, as in Fig. 175, points and d 
coincide and the length of the helix between a and d becomes the normal 
pitch. It is, however, true as before that the normal pitch, multiplied by the 
number of teeth, which is now one, is still equal to the length of the normal 
helix between two intersections with the tooth helix. 

"A glance at Fig. 174 will show that in gears of the same diameter the 
length of the normal helix* grows shorter as the angle kal grows less, and 

* "Length of normal helix" is to be understood as meaning the length of that helix 
between two intersections with the same tooth helix. 



214 



AMERICAN MACHINIST GEAR BOOK 



hence that it and its gear will contain successively fewer and fewer teeth of 
the same normal pitch. That is to say, the number of teeth in a gear varies 
with the helix angle as well as with the diameter, and the number of teeth in 
two gears of the same normal pitch is not necessarily proportional to the diameters. 
In fact, it is never so proportional, except when the angle kal is equal to 45 
degrees. The diametral pitch of the cutters and the diameter of the gear thus 
do not determine the number of teeth. 

"The two facts thus developed are fundamental and will bear restating: 
''First, the number of teeth is equal to the length of the normal helix divided 
by the normal pitch. 

"Second, the numbers of teeth in a pair of gears are not proportional to the 
diameters, except when the angle of the tooth helix w 45 degrees. 



THE SPEED RATIO 



"Fig. 176 illustrates the simplest possible case of a pair of spiral gears. 
The gears are of equal size and the tooth helix has an angle of 45 degrees. 
Such a pair of gears will obviously run at the same speed — that is, have a 




Fig. 179 



Fig. 176 



Fig. 178 



Fig, 177 



THE SPEED RATIO. 



speed ratio of i — and as obviously both will have the same number of teeth. 
Now, unlike spur gears, there are two ways in which the speed ratio of such 
a pair of spiral gears may be varied. First, the diameters of the gears may be 
changed, as with spur gears, the angle of the tooth helix remaining unchanged, 
as in Fig. 177; and second, the angle of the helix may be changed, the diam- 



SPIRAL GEARS 215 

eters of the gears remaining unchanged, as in Fig. 178. These methods act 
in very different ways. The first method is analogous to the procedure with 
spur gears. As with spur gears, the circumferential or pitch-line speed of the 
two gears remains, as before the change, equal, but the length of the circum- 
ference of the two gears is unequal and the largest one thus has a less number 
of revolutions than the smaller one. The second method is entirely unlike 
anything seen in connection with spur gears. By it the pitch-line speeds 
of the two gears are made unequal, and hence, while their diameters are 
equal, the lower one revolves the more slowly. This points out another 
fundamental difference between spiral and spur gears: With spiral gears, 
unless the helix angle is 45 degrees, the pitch-line speeds of two mating gears 
are not the same. 

''The two methods of changing the speed ratio shown in Figs. 177 and 178 
may be combined. That is, part of the desired change in speed may be ob- 
tained by changing the diameters of the gears and the remainder by changing 
the angle of the helix. Given the speed ratio and the diameter of one of the 
gears, we may assume a helix angle and find a diameter for the second gear 
to go with it which shall give the desired speed ratio, and, having done this, 
a second angle may be assumed and a second diameter be found. There are 
thus an indefinite number of combinations of angles and diameters which will 
give the required speed ratio. Note, however, that with the diameter of one 
gear fixed, every change in the diameter of the other changes the distance 
between centers, that not every angle of helix can be obtained by the gears 
which are furnished with universal milling machines, and that if ready-made 
cutters are to be used the lengths of both normal helixes must be exact multiple 
of the normal pitch of the teeth. 

''The limitation of the helix angle is not, however, as serious as is usually 
supposed. The tables for spirals which have heretofore been supplied with 
universal milling machines give but a few of the spirals which can be obtained 
with the change gears which are regularly supplied with the machines. For 
universal milling machines, about two thousand spirals can be cut with these 
gears. 

" Geometrically speaking, there is a wide range of choice in the helix angle. 
As regards the desirabiUty of different angles from the standpoint of dura- 
bility,the conditions are essentially the same as in worm gearing. Reference 
to Charts 10 and 1 1 under worm gears will show that the most favorable angle 
for durability is at about 45 degrees. There is, however, but a trifling 
increase in wear down to 30 degrees, no serious increase down to 20 degrees, 
and no destructive increase down to about 12 degrees. As the angle of worm 
is the complement of the angle of the driving spiral gear, the angle selected 
from Charts 10 and 11, for worm gears, should be the angle of the follower a, 
which is measured from the axis. Where gears are to transmit considerable 
power the best results should attend the use of angles between 30 and 45 
degrees, while angles as low as 20 degrees may be used without hesitation, and 



2i6 AMERICAN MACHINIST GEAR BOOK 

as low as 1 2 degrees if the gears are to run in an oil bath or do light work only. 
The angle may also be increased about 45 degrees by similar amounts and 
with similar results. 

" Fig. 179 is a development of the gears of Fig. 178, the angle a of Fig. 179 
being equal to hat of Fig. 178, but in reversed position, because in Fig. 178 
the upper side of the driver is seen, while in Fig. 179 the direction of the teeth 
is that of the lower side of the driver." 

NOTATION FOR SPIRAL GEARS 

The angle as given for spiral gears is from the axis, which is the opposite 
or complement of the angle for a worm, therefore the angle governing the 
efficiency of spiral gears should be determined from tables on worm gears 
as the angle of the follower {a). 

The greatest angle must always be the driver, except where the angle is 
45 degrees, when either gear may drive. 

All of the tooth parts are derived from the normal pitch. The pitch diam- 
eters are derived from the circular pitch, which is never the same in both 
gears of a pair, except where the angle of both gears is 45 degrees. 

As the diameter of the spiral gear is no indication of its speed ratio, the 
terms gear and pinion are liable to be confusing, therefore follower and driver 
are used. 

Ni = number of teeth in follower. 
iVi = number of teeth in driver. 

di = pitch diameter of follower. 

di = pitch diameter of driver. 

a = angle of follower. 

j8 = angle of driver. 
p'2 = circular pitch of follower. 
p\ = circular pitch of driver. 
p'"* = normal circular pitch (the same in both gears of a pair). 

P = normal diametral pitch (the same in both gears of a pair). 
L2 = lead of follower (length of tooth helix). 
Li = lead of driver (length of tooth helix). 
Z>2 = outside diameter of follower. 
Di = outside diameter of driver. 

s"" = addendum of normal pitch. 

^2 = revolutions of follower. 

ri = revolutions of driver. 
d = angle of shafts. 

C = center distance. 

EXAMPLES 

Specifications for a pair of spiral gears are sometimes given in this manner : 
Required a pair of spiral gears; ratio 3 to i, to operate on 5-inch centers. 



SPIRAL GEARS 



217 







DRIVER 




follower 


REMARKS 




TO FIND 


FORMULA 


TO FIND 


FORMULA 




I 


)8 


d\ ri 
^ P'^ 


a 


90° - 


Axes at right 
angles only. 

Axes at right 


2 


iS 


Tan $ = ., 
p-2 


a 


90° -)8 


angles only. 


3 


i8 


p'n 

Cos i3 = ^, 
pi 


a 


S -fi 




4 


i8 


d\ IT 

Tan fi = r ' 


a 


S -fi 




5 


/>,- 


d\ TT 

Ni '"'^ 


p'n 


(fa IT 

lv7 ''' « 


Same in both 
gears. 


6 


p'n 


p\ cos j8 


P'n 


p\ cos a 


Same in both 
gears. 






p'n 


, f 


p'n 




7 


P'^ 


cos fi 


p'l 


cos a 




8 


P'r 


di TT 

Ni 


P'^ 


d,v 




9 


U 


p'. Ni 


u 


p\ N, 


Axes at right 








angles only. 


10 


u 


d\ ir tan a 


u 


di IT tail )8 




II 


N, 


di P cos )8 


N, 


di P cos a 




12 


N, 


di IT 
P'^ 


N, 


d-i tr 
P'. 






d. 


2C 


d. 


2C 






^^^^ tan aj+ I 




13 


I ^- tan fi \-\- I 

\ / 


Axes at right 
angles only. 






2C 








14 


d. 


/ ri cos /8 \ 

I r2 cos a J ~^ ^ 


d. 


2 C - di 




IS 


dx 


Ni p\ 0.3183 


d. 


N-i p'2 0.3183 




i6 


d. 


N, 


d. 


N, 




P cos iS 


P cos a 




17 


D, 


Ji + 2 5** 


D, 


d-i -\- 2 sn 




18 


D, 


2 
^1 + -p 


D, 


d. + -p 


14?° standard 
only. 




Cutter 


Ni 




_N,_ 




19 


Seecharti4 


cos^ fi 

r ^' 4- 


Cutter 
2 P cos a 


cos^ a 




20 


^ ~ 2 P cos & "^ 








Formulas 


FOR Spiral 


Gears. 





2i8 AMERICAN MACHINIST GEAR BOOK 

The outside diameter of the driven gear must not exceed 7 inches; to be in 
the neighborhood of 6 diametral pitch. 

As the most efficient spiral angle is in the neighborhood of 45 degrees, the 
follower should be made as large as possible, as to obtain this angle the diam- 
eter of both gears must be in proportion to their number of teeth, as for spur 
gears. As the pitch mentioned in connection with spiral gears is always the 
normal pitch, to obtain a trial pitch diameter for the follower twice the 
addendum of the normal pitch subtracted from the outside diameter will give 
the pitch diameter, according to formula 18: 

2 2 

d2 = D2— p = 7— v = 6% inches. 

and, 

fi^i = 5 X 2 - 6% = 3M inches. 

The next step is to find the angle of driver by formula i. 

The angle of follower = 90° — 56°! 9' = 7^3° 41'. 

Find the provisional number of teeth by formula 11. 

Ni = diF COS. = 3^^ X 6 X 0.5546 = 11.092. 
N2 = d^P COS. a = 6% X 6 X 0.8321 = 33.284. 

Naturally the number of teeth must be whole numbers, so it will be 
necessary to change either the center distance, or to make numerous cal- 
culations and shift the diameters. Practically, however, it is possible to have 
quite an error in the normal pitch; the normal pitch, or the pitch of the cutter, 
preferably being under size rather than over. The teeth are thus cut enough 
deeper than standard to secure the proper thickness of tooth at the pitch 
line. 

This difference may be 0.02 of the circular pitch in some cases. 

If the cutter is heavier than the normal pitch it will be impossible to secure 
enough clearance at the bottom of the tooth as the proper thickness of tooth 
will be reached before getting the depth of tooth required. 

If the center distance can be changed the pitch diameters may be shifted 
by the method explained on page 66 of Mr. F. A. Halsey's book — '' Worm and 
Spiral Gearing," as follows: 

final diameter _ 11 

provisional diameter 11.094 
or 



final diameter = provisional diameter X 

That is: 

finalc^i = 2>y^ X — — - = 3-305; 
1 1 .094 



II 



11.094 



SPIRAL GEARS 



219 



and, 



and 



final d^ 



6MX 



33.282 



= 6.610; 



^^ -|- c?2 = 3.305 + 6.610 = 9.915 = twice the corrected center distance. 
In the present example the normal circular pitch for the nearest even 
number of teeth, 11 and ^^^ by formula 5 would be: 






sHx 3.1416 



II 



+ 0.5546 = 0.5280. 



As the pitch of the cutter is 0.5236, this error will not prevent a first-class 
job being turned out if proper precautions are taken, and no change will be 
required in the center distance. 

These points being settled, the remaining calculations are simple. Before 
making any calculations, the requirements should be put in the form of a 
table to avoid confusion, as follows: 



DIMENSIONS 



Pitch diameters 

Revolutions 

Angles 

Number of teeth .... 

Circular pitch 

Normal pitch 

Cutter used 

Lead, exact 

Lead, approximate. . . 

Addendum 

Outside diameter .... 
Whole depth of tooth 
Thickness of tooth . . . 

Gear on worm 

First gear on stud . . . 
Second gear on stud . 
Gear on screw 



DRIVER 


FOLLOWER 


3K 


6K 


56°^9' 


33°'4i' 


II 
0.9520" 
0.5280" 


33 
0.634s" 
0.5280" 


No. 2-6 p 

6.9795" 

6.9670" 
0.1680" 


No.3-6/> 

31.4160" 

31.5000" 

0.1680" 


3.6690" 
0.3630" 

0.2640" 


7.0030" 
0.3630" 
0.2640" 


86 


72 


48 
28 


40 
56 


72 


32 



An error of 0.5 inch in a lead of 50 inches would not ordinarily be pro- 
hibitive, but the angle must be changed to suit any alteration of the lead 
or the cutter will drag. If too much alteration is made in the lead and angle, 
the teeth must be cut a little deeper than standard to allow the gears to 
assemble on the proper shaft angles. 

The amount of adjustment that can be made depends, of course, upon the 
accuracy required, and should be done by some one accustomed to the work. 
This is not possible when cutting helical or herringbone gears, as the tooth 
has contact the entire length of face and a slight error is noticeable. The 
accuracy of the final calculations may be checked by the angles, obtained 
from the circular pitch by Formula 3. 

Another way of presenting this problem is as follows: 



2 20 AMERICAN MACHINIST GEAR BOOK 

Required, a pair of spiral gears; ratio 4 to i; about 8 diametral pitch 
(0.3927-inch circular pitch). Angle of spiral for driver, j8, to be about 55 
degrees, a = go — ^ = 7,$ degrees. 

Find the diameter of driver by formula 13. 

J 2C 2X6 . , 

di = = = 3-1572 mches. 



(~tan.a]-^i I X 0.7002 I + 



The diameter of follower d^ = 2C — di = 12 — 3.1572 = 8.8428 inches. 

The remaining dimensions are found as in the first example. 

Still another example: 

The ratio of a spiral gear drive is 4 to i. The diameter of the driver 
cannot be less than 8 inches, on account of the size of the shaft. The distance 
between centers to be 5^^ inches. No pitch mentioned. 

Assumed diameter of driver di = S inches. 

Diameter of follower f^2 = (5/^^ X 2) — 8 = 3 inches. 

According to formula i : 

Tan, ^ = ^'^ = ^^ = 10.66 or 86° 25'. 
d2ri 3X1 ^ 

Try 7 diametral pitch: 

According to formula 1 1 : 

A^i = diP COS. j(3 = 8 X 10 X 0.0625 = 5 teeth; 
Ni = d^P COS. a = 2 X 10 X 0.9980 = 19.96, say 20 teeth; 
which just happens to come out even. 

If the center distance is not specified, the best plan is to assume number of 
teeth, angles, and pitch of cutter, P or />'" and find the corresponding center 
distance by formula 20. 
Example : 

What center distance will be required for a pair of spiral gears 11 and t^t, 
teeth, 6 diametral pitch, the angle of the 11 tooth drive being 56° 19' and the 
angle of the follower 33° 41'. 

According to formula 20: 

2P COS. ^^ 2P COS. /? 2 X 6 X 0.5546 "^ 2 X 6 X 0.8321 ^' ^^^ 

+ 3.3049 = 4-9578 inches center distance; 

1.6529 being the pitch radius of the pinion, and 3.3049 the pitch radius of the 
gear. 

When the center distance is approximate, this is the simplest solution of 
the problem, the speed ratio being used in place of a trial number of teeth, and 
the number of teeth made to suit the desired center distance. 

A CHART FOR LAYING OUT SPIRAL GEARS 

Chart 13 with the following explanation of its deviation and use will be an 
aid in solving spiral gear problems once the provisional number of teeth 



SPIRAL GEARS 



221 



are obtained. This diagram and explanation were originally published in 
American Machinist, February 27, 1902, by J. N. Le Conte. 

''The provisional numbers of teeth will not in general be whole numbers, 




.5^ 

60 cos a 
^Follower 



CHART 13. DIAGRAM FOR LAYING OUT SPIRAL GEARS. 



1.0 
10°0<' 

2 PC 



but we must choose the nearest whole numbers to the ones obtained, and 
recalculate the angles and radii to fit the new case, as has been previously 
shown. The direct solution of this depends upon the solution of the 
equation : 



+ 



= 2 PC 



COS. a COS. (a — 8) 

''In which Ni and N2 are the nearest whole numbers of teeth to the 
calculated ones, and 8 is the shaft angle or 8 = a -\- ^. As is well known, this 
equation cannot be solved by any simple means, for it is of the fourth degree, 



222 AMERICAN MACHINIST GEAR BOOK 

and, though possible of solution, such solution is not practical. Furthermore 
there are four real values of a which will satisfy it. Graphic methods of 
solution, or continued approximations, must then be resorted to. Chart 13 
gives a method by which the angle can be read off directly. Having obtained 
the nearest whole number of teeth on the gears, find on the diagram the 

point G whose co-ordmates are p^ and p^ on the inner scales. 

Through this point draw a line or merely lay a straight-edge tangent to 
the curve representing the shaft angle. The outer scales on the bottom and 
left will give roughly the angles a. and ^ respectively, and the inner scales the 
values of cos a and cos jS quite accurately. The radial lines of velocity 
ratio will facilitate the location of the desired point, for if the ratio be one 
of those given, the point must lie on its line. It will also be noticed that 
for shafts crossing at 90° the position of the line gives the desired angles at its 
two extremities. 

*'It is interesting to note that two Unes can be drawn through a given 
point tangent to the curves, as shown. As a matter of fact, four such lines 
could be drawn provided the whole of the curves were laid in, but that portion 
shown is the only portion giving positive angles, i.e., angles within the angle 5. 
But there will be two separative positive values of the angle «, which, with 
a given velocity ratio, number of teeth and shaft distance, will work cor- 
rectly together, giving of course different values of the radii. Which of these 
is the one required can always be told as lying nearest to the first approxima- 
tion of the angle. If the point G lies on one of the curves, the two positions 
coincide (a limiting case), and if it lies on the concave side the solution is 
impossible within the angle 5. 

''As an example of the use of the diagram, take the oft-quoted case of Mr. 
De Leeuw. Here angular velocity 
,2 



y\=—, = M, Ni = 8, N2 = 32, C = 4.468", 8 = 90°, and P = 6. 
7 



Then 



= o-i49> ^r^TT = 0.596. 



2PC ^^^' 2PC 



"These co-ordinates give the point G on the diagram. A line through G 
drawn tangent to the lower part of the 90° curve gives quite accurately cos. a 
= 0.894, and COS. ^ = 0.447, o^' 

a = 26° 35' and ^ = 63° 25^ 

agreeing quite closely with the result derived analytically. If the second line 
be drawn through G tangent to the upper portion of the curve, it gives: cos. a 
= 0.787 and COS. ^ = 0.617, or: 

a = 38°6i'andi3 = 5i°54'. 

These fulfill the requirements." 



SPIRAI. GEARS 



223 



SPIRAL GEAR TABLE 



While it is better in every case to understand the principles involved before 
using a table, as this tends to prevent errors, they can be used with good 
results by simply following the directions carefully. The subject of spiral 







To obtain the 




To obtain the 








To obtain the 
circular pitch 
for one tooth 


pitch diame- 
ter, divide by 


To obtain the lead of spiral, 


pitch diame- 
ter.divide by 


To obtain the 






the required 
diametral 


divide by the required dia- 


the required 
diametral 


circular pitch 
for one tooth 






divide by the 


pitch and 


metral pitch and multiply 


pitch and 






required d i - 


multiply the 
quotient by 


the quotient by required 


multiply the 
quotient by 


divide by the 
required dia- 






a m e t r a 1 


the required 




the required 


metral pitch. 






pitch. 


number of 


number of teeth. 


number of 








teeth. 




teeth. 






ANGLE OF 


CIRCUlAR 


ONE TOOTH OR 




ONE TOOTH OR 


CIRCULAR 


ANGLE OF 


SPIRAL 






LEAD OF SPIRALS 






SPIRAL 


DEGREKS 


PITCH 


ADDENDUM 




ADDENDUM 


PITCH 


DEGREES 


Small 


Small 


Small 


Small 


Large 


Large 


Large 


Large 


Wheel. 


Wheel. 


Wheel. 


Wheel. 


Wheel. 


Wheel. 


Wheel. 


Wheel. 


I 


3.1419 


1. 0001 


180.05 


3.1420 


57.298 


180.01 


89 


2 


3-1435 


1.0006 


90.020 


3.1435 


28.653 


90.016 


88 


3 


3-1457 


1. 0013 


60.032 


3.1458 


19.107 


60.026 


87 


4 


3-1491 


1.0024 


45.038 


3.1492 


14.335 


45.035 


86 


5 


3-1535 


1.0038 


37.077 


3.1527 


11.473 


36.044 


8S 


6 


3-1589 


10055 


30.056 


3.1589 


9.5667 


30.055 


84 


7 


3.1652 


1.0075 


25.728 


3.1651 


8.2055 


25.778 


83 


8 


3.1724 


1.0098 


22.573 


3.1724 


7.1852 


22.573 


82 


9 


3.1806 


1.0124 


20.082 


3.1807 


6.3924 


20.082 


81 


10 


3.1900 


1. 01 54 


18.092 


3.1901 


5.7587 


18.092 


80 


II 


3.2003 


1.0187 


16.464 


3.2003 


5.2408 


16.464 


79 


12 


3-2145 


1.0232 


15.076 


3-2I05 


4.8097 


15.104 


78 


13 


3-2242 


1.0263 


13.966 


3-2294 


4.4454 


13.988 


77 


14 


3-2377 


1 .0306 


12.986 


3-2378 


4.1335 


12.986 


76 


IS 


3.2522 


1.0352 


12.138 


3-2524 


3.8637 


12.138 


75 


16 


3.2679 


1.0402 


11.393 


3-2678 


3.6279 


11.397 


74 


17 


3.2848 


1.0456 


10.417 


3.2821 


3.4203 


10.745 


73 


18 


3.3116 


1. 05 14 


10.192 


3-3032 


3.2360 


10.166 


72 


19 


3.3225 


1.0576 


9.6494 


3.3225 


3.0715 


9.6494 


71 


20 


3.3430 


1. 064 1 


9.1848 


3-3433 


2.9238 


9.1854 


70 


21 


3.3650 


1.0711 


8.7662 


3-3652 


2.7904 


8.7663 


69 


22 


3.3882 


1.0785 


8.3862 


3-3833 


2.6694 


8.3862 


68 


23 


3.4127 


1.0863 


8.0399 


3.4129 


2.5593 


8.0403 


67 


24 


3.4451 


1 .0946 


7.7379 


3.4391 


2.4585 


7.7242 


66 


25 


3.4661 


1. 1033 


7.4332 


3.4663 


2.3662 


7.4336 


6S 


26 


3-4953 


1.1126 


7.1664 


3.4952 


2.2811 


7.1663 


64 


27 


3-5258 


1. 1223 


6.9198 


3.5257 


2.2026 


6.9197 


63 


28 


3-5579 


1.1325 


6.6912 


3.5575 


2.1300 


6.6916 


62 


29 


3.5918 


1. 1433 


6.4709 


3.5919 


2.0626 


6.4799 


61 


30 


3.6276 


1. 1547 


6.2778 


3.6277 


2.0000 


6.2832 


60 


31 


3.6650 


1.1666 


6.0979 


3.6652 


1.9416 


6.0997 


59 


32 


3.7043 


1.1791 


5.9282 


3.7044 


1.8870 


5.9282 


58 


33 


3.7457 


1.T923 


S.7710 


3.7459 


1.8360 


5.7680 


57 


34 


3.7894 


1.2062 


5.6181 


3.7826 


1.7882 


5.6178 


56 


35 


3.8349 


1.2207 


5.4754 


3.8351 


1.7434 


5.4770 


55 


36 


3.8830 


1.2360 


5-3431 


3.8834 


1.7013 


5.3448 


54 


37 


3.9336 


1. 2521 


5.2201 


3.9261 


1.6616 


5.2200 


53 


38 


3.9867 


1.2690 


5.1028 


3.9921 


1.6242 


5.1026 


52 


39 


4.0482 


1.2867 


4.9866 


4.0416 


1.5890 


4.9920 


51 


40 


4.1010 


1.3054 


4-8873 


4.1012 


1.5557 


4.8874 


50 


41 


4.1626 


1.3250 


4-7885 


4.1540 


1.5242 


4.7884 


49 


42 


4.2273 


1.3456 


4.6949 


4.2272 


1.4944 


4.6948 


48 


43 


4.2956 


1.3673 


4.6065 


4.2956 


1.4662 


4.6062 


47 


44 


4.367T 


1. 3901 


4.5223 


4.367s 


1.4395 


4.5225 


46 


45 


4.4428 


1.4142 


4.4428 


4.4428 


1.4142 


4.4428 


45 



Table 22 — Spiral Gear Table Shaft Angles 90 Degrees 

For one Diametral Fitch. 

gears is so much more compUcated than other gears that many will prefer to 
depend entirely on tables. 

This table gives the circular pitch and addendum or diametral pitch and 



224 



AMERICAN MACHINIST GEAR BOOK 



lead of spirals for one diametral pitch and with teeth having angles from i 
to 89 degrees to 45 and 45 degrees. For other pitches divide the addendum 
given and the spiral number by the required pitch, and multiply the results 
by the required number of teeth. This will give the pitch diameter and lead 
of spiral for each gear. For the outside diameter add twice the addendum 
of the normal pitch, as in spur gearing. 

Suppose we want a pair of spiral gears with 10 and 80 degree angles, 8 
diametral pitch cutter, with 16 teeth in the small gear, having lo-degree angle 
and 10 teeth in the large gear with its 80-degree angle. 

Find the lo-degree angle of spiral and in the third column find 1.0154. 
Divide by pitch, 8, which is 0.1269. Multiply this by the number of teeth; 
0.1269 X 16 = 2.030 = pitch diameter. Add two addendums or % = 0.25 
inch. Outside diameter = 2.030 + 0.25 = 2.28 inches. 

The lead of spiral for 10 degrees, for small gear, is 18.092. Divide by 

pitch = — '- — = 2.2615. Multiply by number of teeth, 2.2615 X 16 = 

o 

36.18, or lead of spiral, which means that the tooth helix makes one turn in 
36.18 inches. 





CORRESPONDING 


CORRESPONDING 




PITCH OF CUTTER 






ADDENDUM 




CIRCULAR PITCH 


DIAMETRAL PITCH 




P 


P' 


P 


5° 


2 


2.2214 


I.4142 


.70710 


2H 


1-9745 


1.5909 


-62853 


2y2 


1. 7771 


1.7677 


.56568 


2H 


1.6156 


1-9445 


.51426 


3 


1.4809 


2.1213 


.47140 


s'A 


1.2694 


2.4748 


.40406 


4 


1.1107 


2.8284 


•35355 


S 


.8885 


3-5355 


.28284 


6 


.7404 


4.2426 


•23570 


7 


•6347 


4.9497 


,20203 


8 


•5553 


5.6568 


.17677 


9 


.4936 


6.3638 


•15713 


10 


•4443 


7.0710 


.14142 


12 


.3702 


8.4853 


.11785 


14 


•3173 


9.8994 


.10101 


x6 


.2776 


11-3137 


08838 


18 


.2468 


12.7279 


.07856 


20 


.2221 


14.1421 


.07071 


22 


.2019 


15-5563 


.06428 


24 


.1851 


16.9705 


.05892 


26 


.1708 


18.3847 


•05439 


28 


.1586 


19.7990 


-05050 


30 


.1481 


21.2116 


.04714 


32 


.1388 


22.6274 


.04419 


36 


.1234 


25-4558 


.03928 


40 


.1111 


28.2842 


•03535 


48 


.0925 


33-9411 


.02988 



Table 23 — Spiral Gears of 45 Degrees 
For determining the pitch diameters of spiral gears when the pitch of cutter is assumed and 
angle of spiral is 45 degrees: Multiply the addendum of the normal pitch found in fourth colufnn 
by number of teeth. 



SPIRAL GEARS 225 

For the other gear with its 80-degree angle, find the addendum, 5.7587. 
Divide by pitch, 8 = 0.7198. Multiply by number of teeth, 10 = 7.198. 
Add two addendums, or 0.25, gives 7.448 as outside diameter. 

The lead of spiral is 3.1 901. Dividing by pitch, 8 = 0.3988. Multiply 
by number of teeth = 3.988 the lead of spiral. 

When racks are to mesh with spiral gears, divide the number in the circular 
pitch columns for the given angle by the required diametral pitch to find the 
corresponding circular pitch. 

If a rack is required to mesh with 40-degree spiral gear of 8 pitch, look 
for circular pitch opposite 40 and find 4.1 01. Dividing by 8 gives 0.512 as the 
circular pitch for this angle. The greater the angle the greater the circular 
or linear pitch, as can be seen by trying an 80-degree angle. Here the circular 
pitch is 2.261 inches. 

Without the aid of the table, even such a relatively simple problem in 
spiral gear design would entail a complexity of computations which would 
not only be tedious but would be liable to introduce errors which might lead 
to costly shop tim^e wastes and ruined material. For all practical purposes, 
the table will be found extremely valuable and even when odd spiral angles 
have to be employed, necessitating the more laborious calculations, it will 
prove of great assistance in checking the results of the computations. 

DIRECTION OF ROTATION AND THRUST OF SPIRAL GEARS 

The use of spiral gears generally causes some study on the part of the de- 
signer to determine the proper direction of the teeth, having given the direc- 
tion of rotation of the two shafts which are to be connected. Another point 
about spiral gears which also causes some study after the direction of the 
teeth has been determined is the direction of the axial thrust, that is, the 
direction in which the spiral gears tend to move along their axes when trans- 
mitting motion. The proper direction of thrusts is very important to locate 
correctly the ball-thrust bearings of other suitable anti-friction devices. 

It sometimes occurs that a newly designed machine, when started for the 
first time, has a shaft which is driven by spiral gears running in the opposite 
direction to that which was intended, or that the anti-friction washers have 
been located on the wrong side of the helical gear. To obviate this and 
reduce the chance for mistakes in directions of rotation and strains in spiral 
gears, four diagrams, 180 to 183, are arranged to readily illustrate every pos- 
sible combination; giving the direction of the teeth, their rotation, and the 
direction of the lateral strains when they are transmitting motion in the 
directions indicated. These diagrams eliminate the necessity of consulting 
a gear model, nor is it necessary to go through a series of hand manipulations 
describing the rotations in the air. 

In the diagrams. Figs. 180 and 181 represent a pair of right-hand helical 
gears with the direction of rotation of the drivers reversed. 

15 



226 



AMERICAN MACHINIST GEAR BOOK 



The diagrams Figs. 182 and 183 each show a pair of left-hand hehcal 
gears, also with the directions of their drivers reversed. It should be noted 
that reversing the direction of rotation of the drivers reverses the directions 




S 



Briver 

TTTTT^^Tmilh 



FIG. 180. FIG. 181. 

RIGHT-HAND SPIRAL GEARS. 



of their axial thrusts. Also, if the driven gears are made the drivers and 
rotating in the same direction, as shown, the lateral strains are also reversed; 
that is, if in Fig. 180 the driven gear is made the driver and rotates as indi- 




FIG. 182. FIG. 183. 

LEFT-HAND SPIRAL GEARS. 



cated, the gear marked the driver, which is now the driven gear, will rotate 
as shown, but the axial thrust of each gear would be as in Fig. 181. If the 
driven gear of Fig. 181 is made the driver, the lateral strains are as shown in 
Fig. 180. This is also true of the left-hand combinations shown in Figs. 182 
and 183, originally published in the American Machinist by William F. 
Zimmerman. 



SPIRAL GEARS 



227 



h 






s 



o 
o 

t— I 
60 
S3 
<3 

"So 
0? 



^ 



CO 



So 



N, 












— 




... 

N^_ 







^ 


\ 














s 


^ . ... 






\ 
















^ 








\ 














s 










\ 
























\ 


s, 






















N 


\ 






s 

V 


1- - 
















\ 


\ 




1 
s 

s 


m 




s. 
















\ 


, 


Ira 




\ 
















\. 




\, |_ 






\, 














\ 




s 






\ 














\ 




\ 








\, 












\ 1 




In -L 








N 


\, 










^ 


\._._ .S 


N ___ 




\, 






\ 












--- 1 -. 


\ 




\ 








\ 










•1 


^ " 






\ 








\ 








\ a 








\ 










\ 






\ 




\ 


s, 




\ 








s 


\ 




S 






\, 




\ 










\ 




i 




- ^_. 


\ 






\ 










\ ~~i 


P. ._..\, 1 




S 




'\ 




\ 










\ 


4i_.._.. :, 






\, 


\ 






\ 








\ 


Vj :.... 






\ 




\ 






s 






^ 


u 






^ 




\ 






\ 






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^^ 








\, 




\ 






\ 




\ 




us 




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\ 




\^ 






^ 




ife - ^:- 








\, 




s 


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\ 


%- -^-. 




J 




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s, 


\ 




\ 


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\ 


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V, 


\, 




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^■ 


K ^3:__ 


"v T 






s, 


\ 




\ 




\ 




Ji^ \ Si 


_ ^ 


\ 




> \ 




s 


V 




\* 


\ 


^ ^^ ± 


__ ^ 






\^ 




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\ 


% 






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s, 




s, 




\ 


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°, \ -- 




V 




s 






s ''<^ 


s 


i) 




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i , 








V \ 




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\ 




\ 


^ , ^ : 


_ _ .V_ - 


X 






\ \ 


y^- 


\ <> 









^^ \ 










^A M 


U-S-i 


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t^ ^^ 


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\&\ "* 


fev 


tS>. 


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^ 








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N^^ 


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s: \ 


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::::::::::::::: s; 








-o---\ 




\ 


^ J 


i 










\') 


.xf. 




4^ 


-\ 


s: 


s : r :..::.. 


\ 








V 


i 





\ 






SI N " 


V S 










^ 


ft 
5 


= 


^ 


= 


3 N --- 

-^ — l^Vz^ 

i^ y^ — 


mmmm. 


:::::: s 


™ 








— ^-^ 


s, 


s, 


s 




^ — 

^._ ...v.... 


N 


— 5___ 

— -SEE 

\ 










.N 


,N 




S .:^_.. 


-^ :. 


' 


^ 



^ ^' 5;5 rH ?H T-»T-liH (M CJ - »5 c6 ^ >* 25 

Number of Teeth in the Spiral Gear 



_ C3 C3 

OS C3 i-l C<1 



SECTION IX 

Skew Bevel Gears 

Bevel gears which do not have their axes in the same plane, popularly 
known as skew bevel gears, present one of the most complicated construc- 
tions in gearing. Ordinarily, this arrangement consists of a common bevel 
pinion meshing with a bevel gear having teeth of a spiral type (see Fig. 
184), the contact taking place on a plane parallel to that of, but somewhat 
removed from, the one in which lies the axis of the skew bevel gear. 




riG. 184. SKEW BEVEL GEARS. 



The contact action of this combination is somewhat better than that in 
spiral gearing, as the sliding action of the latter is replaced by a combined 
rolling and sliding action. The pitch surfaces of the bevel pinion are frusta 
of a figure generated by the revolution of a straight line about an axis to 
which it is not parallel, a "hyperboloid of revolution. '' 

A typical plan view of a skew bevel gear with its pinion in position is 
shown as Fig. 185, in which the dimension A represents the off set of the pinion. 
The apex point of the pinion lies in the perpendicular axis plane of the gear, to 

228 



SKEW BEVEL GEARS 



2:29 



which point also converge. the profile planes of the gear teeth actually in 
mesh with the pinion. It is obvious then that the profile planes of each 
succeeding tooth, when in mesh, must converge to the same point. This 
results in a circle of apexes for the gear having a radius equal to the offset 
of the pinion — i.e., the succeeding converging tooth profiles, if prolonged, 
would all be tangent to a circle having a diameter equal to twice the offset 
of the pinion. 

The pinion differs in no way from a regular bevel gear and, therefore, 
governs the proportions of the skew bevel gear. If the pinion was not offset, 
the combination simply a set of bevel gears, the pitch diameter of the gear 




An.nACHiNisT 



l/'M \ 



FIG. 185. DIAGRAM OF SKEW BEVEL GEARS. 

would be F-G. The number of teeth in the skew bevel gear is therefore the 
same as would be required for an ordinary bevel gear of such pitch diameter. 
The actual pitch diameter of the skew bevel gear is considerably greater than 
this "equivalent pitch diameter," depending upon the amount which the 
pinion is offset. 

The normal pitch of the gear {B-C) must conform to the circular pitch of 
the pinion, but the circular pitch (D-E) of the gear depends upon its actual 
pitch diameter, the number of teeth being fixed by the pinion and equal to 
the number of teeth required for a common bevel gear of a pitch diameter 
equal to F-G. 

The sliding action of the teeth upon one another also depends upon the 
amount of offset to the pinion shaft. In the combination illustrated in Fig. 
185 it is evident that sliding of the extreme end of the pinion teeth will occur 



230 AMERICAN MACHINIST GEAR BOOK 

from I to D on the gear. This sUding action is accentuated with any increase 
in the dimension A, the Hmiting condition being when A equals the radius 
of the gear, in which case there would be sliding contact only and no turning 
moment. On the other hand, when A equals zero the sliding action is 
eliminated and there is only true rolling contact. 

The principal relationships of the various dimensions, angles, etc., of skew 
bevel gears follow: 

NOTATIONS FOR SKEW BEVEL GEARS 

p = diametral pitch. 
n = number of teeth in pinion. 
N = number of teeth in gear. 
A = offset of pinion shaft. 
6 = angle of offset. 
d' = pitch diameter of pinion. 
D'e = equivalent pitch diameter of gear. 
D' = pitch diameter of gear. 
d = outside diameter of pinion. 
D = outside diameter of gear. 
p' = circular pitch of pinion. 
pi"" = normal pitch of gear. 
p'l = circular pitch of gear. 
El = center angle of pinion. 
Fi = face angle of pinion. 
Ci = cutting angle of pinion. 
£2 = center angle of gear. 
F2 = face angle of gear. 
C2 = cutting angle of gear. 
E'2 = contact angle of gear and pinion = £1. 
J = angle increment. 
K = angle decrement. 
Vi = diameter increment of pinion. 
s = addendum. 

FORMULAS FOR SKEW BEVEL GEARS 

d'= ^ (I) 

D'. = f (.) 

P 

Tan. e = ^ (3) 

D'=^, (4) 

P\ = "^ (5) 



SKEW BEVEL GEARS 231 

Tan. E, = ^ (6) 

Tan. J = ^^^^^ (7) 
n 

Tan. K = ±31^1^:^^ (8) 
n 

F, = E, + J (9) 

C, = E,- K (10) 

d' . . 

s = - (11) 
n 

Vi = s COS. El (12) 

D = D' -{- 2V1, or for greater accuracy = D' -\ jy—^ (13) 

£2 = (90 - ^1) (14) 

C2 = (90 -El) - K (15) 

F. = ^(90-Ei)+JW (^,) 
1^ p. 



DISCUSSION OF FORMULAS 

The formulas for ascertaining the various dimensions and angles of the 
pinion are similar to those for any bevel gear, once the center angle is 
obtained. 

The "equivalent pitch diameter" of the gear is the same as the pitch 
diameter of the regular bevel gear that would give the required speed ratio, 
and is obtained by dividing the number of teeth in the gear by the diametral 
pitch. 

The angle of offset is the angle between the axis plane of the gear and a 
plane passing through the axis of the gear and the common contact point of 
the pitch diameters (outer) of the pinion and gear. Its tangent is obtained 
by dividing the offset of the pinion shaft by half the equivalent pitch 
diameter of the gear, or twice the offset of the pinion shaft divided by 
the equivalent pitch diameter of the gear. 

The pitch diameter of the gear is then obtained by dividing twice the 
pinion shaft offset by the sine of the angle of offset. 

The circular pitch of the gear is equal to the quotient of the pitch circum- 
ference by the number of teeth. 

The tangent of the center angle of the pinion is found by dividing the pitch 
diameter of the pinion by the equivalent pitch diameter of the gear. 

The outside diameter of the gear is usually found by adding to its pitch 
diameter twice the diameter increment of the pinion. This method is not 
quite accurate, as the diameter increment of the gear and pinion is only the 
same when there is no offset to the pinion shaft. The greater this offset is, 
the proportionally smaller does the diameter increment of the gear become. 



232 AMERICAN MACHINIST GEAR BOOK 

A more accurate way of ascertaining the outside diameter of the gear is by 
the use of the second formula. This more accurate method is not absolutely 
correct, however, for it is based on the assumption that the decrease in 
diameter increment is proportional to the ratio of the equivalent pitch diam- 
eter to the pitch diameter of the gear, which is not absolutely true. The 
possible error is so small for any standard gear, however, as to be quite 
immaterial. 

The center angle of each individual gear tooth is equal to the complement 
of the center angle of the pinion, so that the center angle of a skew bevel 
gear may be taken as the complement of the center angle of the pinion with 
which it is to run. 

The cutting angle of the skew bevel gear is likewise the same for each 
individual tooth and is equal to the difference betwen the center angle of the 
gear and the angle decrement. 

The face angle of the skew bevel gear as obtained by the formula given 
is not absolutely accurate, but the error is sufficiently trivial to be safely 
overlooked in practice, unless the face of the pinion is unusually wide and 
the pitch equally small. In such a case, the cut-and-try method of fitting 
the gear to the pinion is advisable, as the calculations involved for an accurate 
mathematical solution are extremely complex. 

The face angle of a skew bevel gear would not be the same as that of a 
bevel gear matched to mate with the skew gear pinion unless the offset of 
the pinion shaft were zero. Such a condition, which would be that existing 
between a set of common bevels of proper proportions, would fix the minimum 
face angle for a skew bevel gear. The maximum face angle would occur with 
the pinion shaft's offset equal to half the pitch diameter of the gear and would 
be one of 90 degrees. Between these limits the face angle of a skew bevel 
gear may be anything, depending upon the difference in the pitch and equiva- 
lent pitch diameters of the gear. Formula (6) is derived on the assump- 
tion that the increase in face angle of the skew bevel gear, from that of a set 
of bevel gears of similar pitch diameters to the condition where there would 
be no rolling action, is governed by the ratio of the pitch diameter of the gear 
to its equivalent pitch diameter. This relationship is only approximately 
accurate, for the actual increase in face angle is not quite constant between 
its minimum and maximum values. For all practical shop requirements, 
however, formula (16) may be considered as correct. Any possible error 
that might arise would but slightly affect the total depth of tooth at the 
small end of the gear where it would be least noticeable or harmful. 

EXAMPLE IN THE DESIGN OF SKEW BEVEL GEARS 

Required a pair of skew bevel gears; 10 diametral pitch, 85 teeth in gear, 
13 teeth in pinion; pinion shaft offset 1}/^ inches. 



SKEW BEVEL GEARS 233 

Pitch diameter of pinion, d' = - = 1.3''. (i) 



10 

o - 

Equivalent pitch diameter of gear, D'e = — = 8.5''. (2) 

Angle of offset, 6, tan. 6 = — ~ — ^ = 0.3529 (3) 

e = 19° 26^ 

Pitch diameter of gear, D' = ^ = o.oi'', say 9.0''. (4) 

0.33271 

Circular pitch of gear, p\ = ^'^^l ^ = o-33''- (s) 

Center angle of pinion, Ei, tan. Ei = -^ (6) 

= 0.1529 
£1 = 8° 42'. 

A 1 . 7- r 2 X O.I';i26 , . 

Angle mcrement, /, tan. J = (7) 

= 0.02327 
/ = 1° 20'. 

All r^ T^ 2.314 X O.I5I26 ,^, 

Angle decrement, K, tan. K = -^-^ — (8) 

= 0.02692 

K=i° 33'- 
Face angle of pinion, Fi = (8° 42') + (1° 20') = 10° 2'. (9) 
Cutting angle of pinion, Ci = (8° 42') — (1° 33) = 7° 9'. (10) 

12 
Addendum, s = ^— = o.i". (11) 

13 

Diameter increment of pinion, Vi = o.i X 0.1513 = 0.01513''. (12) 
Outside diameter of gear, Z> = 9 + 2 X 0.01513 1 

or Z) = 9 + ^ X 0.01513 X 8.5 = 9.03''. (13) 

Center angle of gear, £2 = (90 — 8° 42') = 81° 18'. (14) 

Cutting angle of gear, Co = (90 - 8° 42') - 1° 33' = 79° 45'. (15) 

TT ^ f 17 [(9o-8°420 + i°2o n 9 00,,^, 
Face angle of gear, F2 = ^^^ ^^ — =87 29'. (16) 

MACHINING SKEW BEVEL GEARS 

Any of the machines used for cutting the ordinary type of bevel gear can 
be used for machining skew bevel gears, if simple adjustments or modifica- 
tions are made. The carrying spindle of the machine must be offset from the 
plane of the cutting tool by a distance equal to the offset of the pinion shaft. 
The subsequent operations of cutting the teeth are similar to those employed 
in cutting plain bevel gears, the rotary adjustment of the gear being governed 
by the circular pitch of the gear, not its normal pitch which corresponds to 



234 



AMERICAN MACHINIST GEAR BOOK 



the circular pitch of the pinion. The adjustments are somewhat more com- 
pHcated than when cutting the simpler gears and must be performed with 
great care, as there is no common apex toward which to work. This adds to 
the difficulties of accurate workmanship and is the main reason why skew 
bevel gears are so seldom employed. 

Another method of designing and cutting skew bevel gears that is some- 
times employed is to make both the teeth of the gear and the pinion of 
spiral type. When this is done, the degree of obliquity of the teeth in the 
gear and in the pinion is made the same in order to facilitate manufacture 
and design. A layout for such gear is shown in Fig. i86. 




American Maehinilt 



FIG. l86. LAYOUT FOR SKEW BEVEL GEARS. 



These gears are turned up according to the dimensions for bevel gears of 
the same number of teeth, pitch, and ratio and no alteration in the diameters 
is usually made or is any alteration in the angles necessary, due to the fact 
that though the apex points of the two gears do not coincide, the converging 
conical surfaces are parallel to those of bevel gears with a common apex 
point. 

Both gear and pinion are machined with the plane of the cutting tool offset 
from the carrying spindle of the machine. This offset is different for the 
two gears if their speed ratio is other than i to i. For gears of similar d'men- 
sions, the total offset of the shafts would be divided in two and the correct 
offset between the spindle of the machine and the cutting tool plane would 
be one-half the total offset for both gears. For any other speed ratio, the 
total offset is divided proportionally to the ratio, the smaller offset being 
employed for cutting the pinion and the larger for machining the gear. For 
instance, when cutting skew bevel gears having a shaft offset of 2 inches 



SKEW BEVEL GEARS 235 

and a speed ratio of 2 to i, the machine drop or offset for the pinion would 

2 
be — ^T^ — = 0.666 inch and for the gear, 0.666 X 2 = 1.333 inches. 
2 I i 

Skew bevel gears cut according to this apportioning method have proved 
very satisfactory, and the only criticism that can be advanced is on account 
of the decreased strength of the teeth as the gears are usually cut. The 
teeth being inclined to the circumference of the gear — that is, not being radial 
— the circular pitch must necessarily be greater than that of common bevel 
gears of similar proportions, for the circular pitch of the common bevels 
corresponds to the normal pitch of the skew bevel gears. This would neces- 
sitate an increase in diameters, the amount of increase depending upon the 
angularity of the teeth. If this is attended to, the full strength of the teeth 
will be developed. 

The obliquity of the teeth of skew bevel gears of all varieties is the cause 
of one other annoyance, due to the unavoidable sliding action between the 
teeth. It has been found that if the common 143^^-degree involute tooth is 
used the teeth do not clear properly. This has been overcome by making the 
angle 20 degrees. In extreme cases an even greater angle might have to be 
employed, but for any ordinary installation of skew bevel gears, the adoption 
of the 20-degree involute tooth will allow the teeth to clear satisfactorily. 



SECTION X 

Intermittent Gears 

Intermittent gears are designed to allow the driven gear or follower one or 
more periods of rest during each revolution of the driver. This may be 
accomplished in a rough manner by cutting out a number of teeth in the fol- 
lower as illustrated in Fig. 192, but the cut and try method must be employed 
to obtain a definite ratio. This type of intermittent gear is seldom used 
there being nothing but the spring b to keep the follower from moving during 
a period of rest, and the first tooth of the driver enters contact in a very 
uncertain manner, it sometimes being necessary to shorten the first tooth 
in the driver to prevent it from striking the top of the first tooth in the 
follower. 

The proper design of intermittent gears is not as difficult as it first appears. 
The pitch and outside diameters are found as for an ordinary spur gear, the 
pitch desired must correspond to an even number of teeth. The blank space 
on the driving gear is milled to the pitch line, and the stops in the follower 
are cut by a cutter of a diameter corresponding to the pitch diameter of the 
driver. If no such cutter is at hand, use the nearest to that size to rough 
out the stops and finish them with a fly cutter which can be set to any desired 
radius. 

It is well not to have the gears too near the same size; the driver should be 
the smallest in order to secure all the contact possible in the stops. 

The simplest form of intermittent gear is shown by Fig. 193, the follower 
being moved but a short distance for each revolution of the driver. It will 
be noticed that a small amount of fitting will always be required at the point 
a to allow the point of the stop to clear. 

A more complicated drive is shown in Fig. 194, the follower being moved 
one-sixth of a revolution for each revolution of the driver. Each of these 
gears is turned up as for a spur gear of 30 teeth 5 diametral pitch. The cut- 
ting operation would be as follows: Index for 30 teeth; cut the first three teeth, 
then index for two teeth without cutting, and so on around the blank. The 
six stops are then milled, with a cutter 6 inches in diameter, to a depth of 
0.2 inch, or the addendum of the gear, which completes the follower. Four 
teeth are then cut in the driver, and the remainder of the blank milled to the 
pitch line. A little filing and the gears are complete. 

A still simpler method of cutting these gears, and one that avoids the 
necessity of first laying them out, is as follows: Drop a cutter equaling the 

236 



INTERMITTENT GEARS 



m 



pitch diameter of the driver into the blank of the follower, to the depth of 
the addendum at the points stops are desired. Then cut the first tooth at a 
point midway between two of the stops, and continue cutting toward one 
of the stops until the point of the stop touching the outside circumference 




American Machinist 



FIG. 192. POOR DESIGN OF INTERMITTENT 
GEARS. 





Anusrican MacfiinUi 
FIG. 193. 
INTERMITTENT GEARS WITH TWELVE STOPS. 



American Sfachinitt 



FIG. 194. INTERMITTENT GEARS WITH 
SIX STOPS. 



of the blank is cut away, or, in other words, until there is no blank space on 
the gear between the last space cut and the point of the stop. The same 
number of teeth are then cut in the opposite direction until the same 
condition is met. If the stops are evenly spaced the cutting of the remaining 
teeth is a simple matter. If the stops are not evenly spaced the first tooth 
for each group must be located between each stop. The same number of 
teeth are then cut in the driver as there are spaces in the follower for each 
group and the remainder of the blank milled to the pitch line. For a pair 



238 



AMERICAN MACHINIST GEAR BOOK 




American Machinist 



FIG. 195, INTERMITTENT BEVEL GEARS. 



of gears such as snown in Figs. 193 and 194, the cutting of the teeth by this 
process will be a simple matter. 

The cutting of internal intermittent 

gears is a counterpart of the above. 

Bevel gears, while being more difficult 

to cut, are governed by the same rules 

(see Fig. 195). 

MODIFICATIONS OF THE GENEVA STOP 

The accompanying engravings illus- 
trate three highly ingenious and extremely 
interesting modifications of the device 
used in watches to prevent overwinding 
which have been applied by Mr. Hugo 
Bilgram to various automatic machines constructed at his works. The 
constructions have a family resemblance in principle, though they are entirely 
unlike from a structural standpoint. 

Three main features characterize the constructions: First, the intermittent 
motion of the Geneva stop; second, the entire absence of shock at engagement 
or disengagement; and third, the positive character of the movement — the 
parts being locked in position both when they are in motion and when they 
are idle. 

Fig. 196 represents the smallest departure from the watch mechanism. 
The interrupted disk a is the driver and revolves continually. The driven 
piece is seen at b, and the requirements are that the driven piece shall remain 
at rest during three-fourths of a revolution of the driver, and shall then make 
one-quarter of a turn during the remaining quarter turn of the driver. The 
driver may revolve in either direction, but supposing it to turn in the direction 
of the arrow, a roller c attached to the driver is about to enter one of four 
radial slots in the face of the driven piece. During the succeeding quarter 
turn of the driver the parts will move together, the motion of the follower 
ceasing when groove d has reached the position occupied in the figure by 
groove e, and this movement of the follower will obviously occupy 90 
degrees of angle. It will be seen that the parts are so laid out that roller c 
enters and leaves the grooves tangentially, insuring absence of shock at both 
the commencement and the conclusion of the engagement. 

At fghi on the follower is a series of rollers raised above the faces sur- 
rounding the grooves, and the circular part of the driving disk carries a circular 
groove jkl at such a radial distance as to engage these rollers in succes- 
sion during the idle period of the follower and hence lock it in position. It is 
obvious that in the direction of motion supposed, this circular groove is just 
leaving roller i, and so disengaging it preparatory to movement by roller c. 
On the completion of the follower's movement, roller/ will occupy the position 



INTERMITTENT GEARS 



239 



of roller ^ in the illustration, while the end / of groove jkl will have turned 
to a position ready to embrace it and so lock the follower in position. With 
motion in the opposite direction, groove jkl in the position shown would be 
in the act of engaging roller i on the completion of the movement. Rollers 
g and i being at the same distance from the center m of the driver, both are 
engaged by the groove during a revolution, the locking taking place with one 
and the unlocking with the other, both rollers being in the groove during most 
of the time. 




FIG. 196. GENEVA STOP — MODIFICATION NO. I. 



A modification of this gear, which it is unnecessary to show, has five 
grooves in the driven wheel with corresponding modifications in the character 
of the movement. 

In the second construction the motion of the follower is intermittent like 
the last, but with different relations between the idle and acting periods. 
The driver runs continuously, the relationship being: 

During % turn of the driver the follower is at rest. 

During i3^^ turn of the driver the follower makes one complete turn. 

In other words, the follower makes one turn to every two turns of the 
driver, but this revolution of the follower occupies little more than a turn of 
the driver. 

16 



240 



AMERICAN MACHINIST GEAR BOOK 



Pinned to the face of the driven gear is the plate a, Fig. 197, the arm b of 
which is fitted to embrace a hub /on the driver shaft, whereby, until released, 
the follower is locked in its idle position. Revolving with this hub is an arm 
c carrying a roller d, which is fitted to engage the slot g. As it does so, the 
notch e in hub/ comes opposite finger h, thus disengaging the locking mechan- 
ism. Roller d enters tangentially without shock and accelerates the motion 
of the follower until the roller reaches the line of centers, when the gears engage 

and the motion goes on. The comple- 
tion of the revolution of the driver finds 
the roller d again on the line of centers, 
but engaging slot i, and the continuance 
of the motion brings the parts again to 
the positions shown. It should be noted 
that in this mechanism not only is the 
starting and stopping of the driver with- 
out shock, but at the instant of engage- 
ment of the gear teeth the roller d has 
brought the velocity of the follower up 
to that due to the gears, so that the trans- 
ition of the motion from the pin to the 
gears and back again from the gears to the 
pin is also without shock. 

The most elaborate of these mechan- 
isms is that shown in Figs. 198, 199, and 
200. In this the driver — turning about 
a — is required to turn through about 73 
per cent, of a revolution, while the follower 
stands still, the follower then making a 
complete revolution during the remaining 27 per cent, of the revolution of 
the driver. Figs. 198, 199, and 200 are side views intended to show the 
action in a succession of positions. 

The driver is an interrupted disk bed, Fig. 198, having cam-shaped 
edges ef at the mouth of the notch. Slightly in the rear of this disk is a 
toothed sector g. The incomplete driven gear h meshes with g during the 
acting periods, and the purpose of the remainder of the mechanism is to start 
b in motion and throw the teeth in mesh as well as to lock the follower in posi- 
tion during the period that it stands still. Fig. 1 98 shows the parts in position 
at the beginning of the movement of the follower, which is still locked in the 
position which it occupies during its idle period. A bar i on the front face 
of gear h rides on, and up to this point has been locked in the idle position 
by the disk bed. Finger / — one of a pair jk — is attached to the rear 
of gear h, where it may turn freely between the sector g and the driving 
pulley I. This pulley / carries two rollers mn arranged to engage the 
fingers/^ respectively. In the position of Fig. 198 the driving disk, moving 




FIG. 197. 



GENEVA STOP — MODIFICA- 
TION NO. 2. 



INTERMITTENT GEARS 



241 





FIG. 198. GENEVA STOP — MODIFICATION NO. 3. 




FIG. 199. FIG. 200. 

GENEVA STOP — MODIFICATION NO. 3. 



242 



AMERICAN MACHINIST GEAR BOOK 



in the direction of the arrow, has brought roller m into position, where it is 
about to engage finger j, the direction of the acting side oi j being tangential 
to the motion of m, so that the movement begins without shock. To permit 
j, b, and i to turn, the edge e of the disk is dressed off, but to such a degree 
that contact is maintained between the right-hand end of i and the edge 
of the disk as the follower turns, so that the motion is positive without slack. 
As the movement progresses the speed on the follower increases until the posi- 
tion of Fig. 199 is reached, when — the roller being on the pitch circle of sector 
g — the speed of the follower is the same as that due to the gears and the teeth 
drop into mesh without shock. From this on the gears drive, and the arms 
k turn completely over, the position when the gears go out of mesh being 




PIG. 201. 



FIG. 202. 
AN INTERMITTENT SPUR GEAR. 



FIG. 203. 



shown in Fig. 200. From this on the action is the reverse of that shown by 
Figs. 198 and 200, the driving piece being now the cam-shaped edge/ of the 
disk, the finger k preserving the positiveness of the motion and preventing 
the driven pieces overrunning by momentum as they are brought to rest, the 
final stopping being accomplished as the roller slips out of action in a tan- 
gential direction, and again without shock. From the position of Fig. 200 to 
that of Fig. 198 the bar i simply rides on the edge of the driver disk and the 
follower remains at rest. 

''It should be remembered that these mechanisms are not models designed 
to embody a pretty movement invented beforehand, but they are parts of 
machines, some of which are made in considerable numbers, and have been 
devised, as occasion arose, to accomplish certain required results. As such, 
they represent the art of invention carried to a high degree of perfection." 

The line cuts (Figs. 201, 202, and 203) show a pair of intermittent spur 
gears in three positions. The peculiarity about this gear is that although a 
dwell occurs, the teeth of the gear and pinion are in mesh at all times. 

The mutilated gear A is the driver and is secured to its shaft. A portion 
of its rim — dependent upon the length of dwell — is cut away. A segment 
B is mounted on the same shaft. This segment is free to swing in the cut- 
away portion of A , and is held in place against the side of ^ by a collar on the 
shaft. The teeth of B match with the teeth of A in both of its extreme posi- 




INTERMITTENT GEARS 243 

tions. B is held against the face C by a spring X. This spring is elastic 
enough to allow B to move as far as D. 

As the gear A moves in the direction of the arrow, it turns the pinion E. 
When B engages with E — the resistance of E being greater than the resistance 
of the spring X — the segment B remains stationary, while A moves till D 
comes in contact with B. During the time that B is at rest the pinion E is 
of course also at rest. As soon as D 
comes in contact with B, both B and E 
begin again to move. When B reaches 
a position where its teeth are no longer 
in engagement with the teeth of E, the 
spring X returns it to the face C. These 
gears were used as a feed gear for paper, 
the paper being cut during the dwell. fig. 204. an intermittent worm. 

The half-tone, Fig. 204, shows three 
views of an intermittent worm used in a looping machine. The pitch is 
1-6 inch. The dwell is two- thirds of a turn, and the advance the remain- 
ing third. It was cut on an ordinary 16-inch lathe, using a mutilated 
change-gear. 

AN INTERESTING PAIR OF SPIRAL INTERMITTENT GEARS 

Figs. 205 and 206 show a pair of intermittent gears having the peculiar 
characteristics that, if the large gear be rotated continuously in one direction, 
the pinion will rotate alternately three-quarters of a turn in one direction and 
one-quarter of a turn in the opposite direction, with a rest or dwell between 
each movement. Similar gears have been made as part of a certain machine, 
the nature of which we are not at liberty to mention. The angle of spiral of 
both gears is 45 degrees, and under ordinary conditions either gear could 
be the driver. The large gear, however, has around portions of its periphery 
two tongues — which are practically a continuation of certain of the teeth. 
These tongues fit into grooves in the pinion, and during their passage 
through the grooves lock the pinion at rest. Owing to this feature, the large 
gear must in this case be the driver. 

The pinion has twelve teeth, divided into four groups of three teeth each, 
with one of the before-mentioned slots between each group. Two opposite 
groups of teeth are cut left hand, the two alternate groups are cut both left 
and right, leaving the teeth like a series of pegs. 

Imagine the handle at a position opposite the pinion, then the left-hand 
teeth in the large gear will be at the left. There are nine teeth cut in this 
segment which, when the handle is turned — in the direction of the hands of a 
clock — engage first with the three teeth of a double-cut group on the pinion, 
then with the three left-hand full teeth of the next group, then with the three 
teeth of the other double-cut group. The pinion has then turned three- 
quarters of a revolution. The tongue then engages with the slot in the pinion 



244 



AMERICAN MACHINIST GEAR BOOK 



and the rotation of the pinion is arrested. The large gear turns until the three 
right-hand teeth on its periphery come into mesh with the double-cut group 
first referred to, and reverse the direction of rotation of the pinion for a space 
of three teeth, or one-quarter turn, when the tongue again locks the pinion 
and the handle reaches the starting position. The large gear has thus made one 
turn and the pinion has advanced through three groups of three teeth each, 
equal to three-quarters of a turn, and has reversed through one group of 
three teeth, or one-quarter turn. Thus the total advance is but two groups 
of three teeth, or one-half turn, and to bring the gear and pinion into the 




FIG. 205. THE RIGHT-HAND TEETH IN 
MESH WITH THE ''PEG" SEGMENT. 



FIG. 206. 



THE LEFT-HAND TEETH 
IN MESH. 



same relative position as they were at the start the wheel must make another 
complete turn. 

The blank for the large gear was turned to the extreme diameter across the 
top of tongue, mounted in the milling machine, and the left-hand teeth, which 
extend clear across the face, were gashed slightly below the level of the top 
of the tongue. The blank was then indexed halfway around from the 
central left-hand space, the table swung for right hand, and a deep right-hand 
gash made for locating. Then the blank was indexed halfway around from 
the central right-hand space, the table swung back for left hand, and the 
left-hand teeth section milled down to the proper gear diameter. Then the 
full-length left-hand teeth were cut, and also a gash made outside of the 



INTERMITTENT GEARS 



245 



end teeth which join the tongue so as to get the curve of the tooth, but not 
going far enough to touch the tongue. The blank was then indexed back 
to the right-hand locating space; the table swung, blank milled down to the 
proper gear diameter, and the right-hand teeth were cut the same as the left- 
hand. The table was then set square, and an end mill was put into the 
spindle, and the stock milled away — leaving the tongue. It will be noticed 
that it was not necessary to remount the blank at all. The outsides of the 
teeth joining the tongue were chipped out as smooth and true to curve as 
possible. 

An important feature was to get the angle of the end of the tongue just 
right, and not file it too far back, which would have allowed the large gear 



^ 





FIG. 207. LAYOUT OF A PAIR OF INTERMITTENT SPIRAL GEARS. 



to carry the small one somewhat beyond the point where the tongue should 
enter the slot, thus causing the end of the tongue to strike the side of the 
small gear and arrest the movement or perhaps cause breakage. The cutting 
of the small gear was simply a matter of setting the cutter directly over the 
center of the blank and swinging the table for both hands; the slots were 
cut at the last machine operation, and then the little sharp projections which 
were left at the ends of the crosscut sections and near the slots were chipped 
off, as they were surplus and would interfere with the movement of the gears. 

Fig. 207 shows layout of a set of the same style of gearing as Figs. 205 and 
206, but of a different ratio. 

The gears were cut by the Boston Gear Works, and were prepared pre- 
liminary to cutting into more expensive blanks. The work was accomplished 
by the usual methods, and the finished gearing accomplished the desired 
results and is now in successful use. 



SECTION XI 

Elliptical Gears 

Elliptic gears are in general use on shapers, planers, slotters and similar 
machine tools to transmit a quick-return motion to the ram. The cost of 
production is more than for gears having circular pitch lines, but they are 
undoubtedly the cheapest quick-return motion among known mechanical 
movements. The method to be outlined is not new, but is as accurate as any 
and the cost of the tools is not high. This method is in general use in many 
shops to the exclusion of other methods not considered as good. 

In order more clearly to describe the process it will be well to take an 
actual case and carry it through from start to finish. Assume that a pair of 
gears is ordered to transmit a 3 to i quick-return motion and the centers are 8 
inches apart. About 3 diametral pitch is specified. 



.--— ir-?. 



METHOD OF LAYING OUT^ 

Fig. 208 represents diagrammatically the method of laying out the gears. 
Lines A A' and BB' are first drawn perpendicular to each other. The major 

axis of the pitch line of the gear is the 
same as the given center distance, 8 inches, 
and is laid off as shown. The focus 
points XX' are drawn in so that ^X is in 
the same ratio to A 'X as the given quick- 
return ratio. The points A and B' are 
located by setting the dividers at one- 
half the major axis BA' and cutting the 
minor axis with arcs having centers at X 
and X\ Arcs having radii equal to OB' 
and OA' are drawn covering one qua- 
drant, as shown. This quadrant is divided 
into six equal divisions, as shown by radii 
from the center O. These radii are 
marked a, a^, a^, etc. Next intersecting lines are drawn from the intersec- 
tion points of a, a^, a^, etc., with the circles, as shown, and the intersection 
points of these perpendiculars are points on a perfect ellipse. 

^'With a center on AA'^ find an arc that will very nearly cut the points 
from a^, a^, a^, and A ^ Also find a center on BB^ about which an arc can be 
drawn cutting points B^, a, a\ and a^. These centers are used when cutting 

* W. E. Thompson. 

246 




FIG. 208. 



LAYOUT FOR AN ELLIPTIC 
GEAR. 



ELLIPTICAL GEARS 



247 



the teeth and should be laid out as accurately as possible. After finding two 
centers and measuring the respective distances from O, the other two centers 
may be put in and an ellipse drawn, as shown. This line is then stepped off 
with dividers set at the corrected tooth thickness for the desired pitch and the 
number of divisions noted. 

'' It is preferable to have an odd number of teeth for convenience in cutting, 
so if the pitch line does not divide into an even number of divisions, half of 
which is an odd number, when the dividers are set to the chordal-tooth thick- 
ness of the desired pitch, the pitch line may be reduced or enlarged, as is 
found necessary. When the pitch is given, as it was in this case, the pitch 
line may be divided and the divider division measured. The corresponding 




. rm . 



1 fl 



A.rt»or 



Vernier 
O M Vernier V 



1 r 



-1 r 



Drive Pin 



A 



"^nfaBvie 




^^ 






op®! 



t -f-; 



— *-i — t--i — t^.^:^ 

Hi 1(31 ~~ '"■!?) ~ ' 

a il 'M l ■ ' 



'^ 



End Elevation 



Side Elevation 



mr 



Plan. 



FIG. 209. FIXTURE FOR CUTTING ELLIPTICAL GEARS. 



pitch is used in selecting a cutter. In this case the line was divided into 30 
divisions which corresponded very close to 2^ pitch, so these cutters were 
used. 

•'Radial lines common to two centers of the elliptic arcs are drawn, as 
shown at 6, i\ 5^, and in the other quadrant not drawn. These lines are the 
dividing points between two different arcs and are drawn before laying out or 
cutting the teeth. Radial lines from the four centers are drawn, through the 
centers of the space divisions, as shown at c, c\ c^, etc. These lines are for 
the purpose of starting the first cut and checking the rest to prevent large 



errors 



The base circles of the two different curves are drawn, as shown, and a 
few teeth laid out by an accurate odontograph on each curve, as shown, or 
cutters may be selected by measuring the pitch radii DA and EB and 
figuring the proper number of teeth to cut. Cutters are selected by one or 
the other of these methods and the gear, previously having the shaft and 
driving pin holes X and X^ bored and the addendum outline roughed, is ready 
for cutting. 

METHOD OF CUTTING 

''The fixture shown at Figs. 209 to 211 is bolted onto a circular milling 
attachment. The fixture consists of a base plate carrying two slides moving 
at right angles to each other. These slides are equipped with verniers regis- 



248 



AMERICAN MACHINIST GEAR BOOK 



tering zero when the center of the arbor is in line with the center of rotation of 
the mining attachment. The drive pin is riveted into a separate sHde that is 
adjustable and is in Hne with the arbor and Hne of motion of the large slide. 
This fixture is placed in position on the machine and the rough-gear blank 
clamped in place. 

The point D^, Fig. 208, is then set over the center of rotation of the attach- 
ment by means of the verniers and the center of the cutter brought into the 
line AA'^. In this case the cutter is an end mill used to finish the outside of 




FIG, 210. ELLIPTICAL GEARS WITH BORE IN CENTER, SHOWING SEPARATION OF TRUE 

ELLIPTICAL PITCH LINES. 



the gear. By moving around each quadrant setting the corresponding centers 
over the center of rotation for each quadrant the outside is milled true with 
the pitch line. A cutter for the ends is then put in the machine and in line 
with AA^. The first space may then be cut to depth with two or more cuts, 
depending on the tests for alignment. A piece of glass having a center line 
and a number of short lines scratched equal distances apart and from the 
center line is used for testing the cuts. The center line is placed over the line 
^^^ on the first cut, and the short lines brought intersecting the pitch line. 
By using a glass a variation of o.ooi inch may be readily seen and remedied. 
After the first space is cut the gear-tooth calipers may be set and used to check 
the cutting. By checking with both the radial center lines and the tooth 
calipers errors are reduced to a minimum. After the first gear is cut it may be 
used as a templet for all others, or the indexing may be noted and repeated." 
Several years ago the author had an experience cutting elliptical gear that 
will be of interest. The driven gear was required to have two variations of 
speed per revolution; therefore the bore was put in the center of the gears 



ELLIPTICAL GEARS 



249 



instead of at the foci, which is usual. The gears cut are shown in Fig. 214. 
When the elhpse is very fiat the four-arc method, described above, cannot be 
employed, as the pitch line cannot be described even approximately correct 
by four arcs. Therefore the gears were cut as illustrated by Fig. 211, the 
blank being set for each tooth cut. The teeth were first located on a templet 
which was secured to the face of one of the gears; they were both cut at one 




FIG. 211. CUTTING THE TEETH. 



time. The blank was manipulated until the center Hne of each tooth space 
was brought in line with the center line of the cutter. 

This is accomplished with a surface gauge placed against the front ways of 
the milling machine as illustrated in Fig. 211. The depth of the teeth was 
obtained by first bringing the cutter to the outside of the templet and raising 
the table a distance equaling the depth of the tooth, or by locating the pitch 
line of the templet with another surface gauge from the cutter spindle. 

A cutter made to finish the outside diameter as the teeth are cut is a 
decided improvement over first milling or slotting the outer surface, although 



250 



AMERICAN MACHINIST GEAR BOOK 



milling off the points of the teeth after they are cut is the next best plan. 
When the gears were mounted as shown in Fig. 210, it was found that the 
pitch lines separated at four points of the ellipse, as between h and c, therefore 
there was excessive backlash between the teeth at these points. 





FIG. 212. THE GARDENERS ELLIPSE. 



FIG. 213. THE METHOD EMPLOYED IN LAY- 
ING OUT GEARS SHOWN IN FIGS. 214 
AND 215. 



The gears had been laid out with a gardeners' ellipse, using a piece of silk 
thread looped around pins set in the foci; the loop being adjusted until the 
describing point passed through the intersections of both the major and minor 




FIG. 214. ELLIPTICAL GEARS LAID OUT AS SHOWN IN FIG. 213. 

axes as shown in Fig. 212. Of course, we all thought the gardeners' ellipse 
was at fault, and it was decided to employ this process only on "tulip 
patches" in the future. 

A proper instrument was secured for the next attempt and everything done 
according to Hoyle. This time, instead of first cutting the gears and inves- 



ELLIPTICAL GEARS 251 

tigating afterward, two templets were laid out to represent the pitch lines 
of the gears. These templets developed the same error found in the first 
gears, so the pitch Unes were corrected as per dotted lines in Fig. 210 and the 
gears were satisfactory. 

Later on several gears of the same size were required and a search was 
made for a simpler method of laying them out. Noticing the attachment 




FIG. 215. INTERFERENCE OF GEARS SHOWN IN FIG. 214, WHEN BORE IS AT FOCI. 

described by Mr. George B. Grant in his ''Treatise on Gears," section 150, an 
attempt was made to apply this principle. A circle whose radius equalled 
the radii of both the major and minor axes was first drawn and divided into 
the number of teeth to be cut in the gear. The ellipse was then described from 
the same center. The center lines of the teeth were then projected from the 
points located on the outer circle to the ellipse, the edge of the blade being on 
a line with the points e and / as shown in Fig. 213. The center line of the tooth 
thus projected did not always cross the pitch line at right angles, except at 
the major and minor axes, and doubts were expressed as to the success of 
gears cut on these lines, but it was thought worth a trial at least. The pitch 
lines were corrected in the same manner as before and the gears were cut. 
They not only ran better than the first gears, but it was found necessary to 



252 AMERICAN MACHINIST GEAR BOOK 

remove part of the correction between the points b and c. This is accounted 
for by the fact that the circular thickness of the teeth increased between these 
points owing to the obhquity of the teeth; the thickness of the tooth being 
measured on the normal section (see Fig. 214). 

Gears with the bore located at the foci will not operate when cut in this 
manner; the center lines of the teeth must be at right angles to the pitch line 
at all points. The interference of the teeth when thus mounted is shown in 
Fig. 215. 

When the bore is in the center of an elliptical gear it is better to have an 
even number of teeth, otherwise the gears must be cut separately. For an 
odd number of teeth the tooth centered on the major axis must engage a space 
located at the minor axis of the engaging gear. It is better to use an even 
number and place the edge of the teeth on the major and minor axes, for gears 
of this type. 

For a complete treatment of the subject of elliptical gears Grant's "Trea- 
tise on Gear Wheels" is to be recommended. 



SECTION XII 

Epicyclic Gear Trains 

CALCULATIONS RESPECTING EPICYCLIC GEAR TRAINS 

DERIVATION OF FORMULAS FOR SEVERAL USUAL TYPES, AND EXTENSION OF THE 
METHOD OF ANALYSIS TO A SOMEWHAT COMPLEX EPICYCLIC TRAIN* 

This form of gearing, which is really that in which one gear revolves around 
the center of one with which it is in contact, has received considerable at- 
tention, and one notices its use in several directions. We will therefore look 
into some of the calculations respecting it, leading from the simpler to the 
more complex. 

SIMPLE PAIR OF GEARS IN FIXED BEARINGS 

Example I. 

If in Fig. 216 R and N are two gears in mesh, r and n being their respective 
numbers of teeth, their bearings being fixed, then: 

Velocity of drive N gear N _ r 

Velocity of drive R gear R n ' 
or, 

f 

N^s velocity = R's velocity X -* 

If, however, R revolves in a positive direction, n must revolve in the opposite, 
that is, in a negative direction. 

r 

.'. N's velocity = R^s velocity X. -• (i) 

n 

In all these calculations it is essential that great care be taken in order to 
obtain the correct sign of the resulting velocity. 

GEARS IN FIXED BEARINGS, WITH AN IDLER 

Example II. 

An intermediate gear / is placed in contact with both TV and R, Fig. 217. 
The effect will be that of giving N motion in the same direction as R. 

y 

.'. N^s velocity = R^s velocity X -• (2) 

* Francis J. Bostock. 
253 



254 



AMERICAN MACHINIST GEAR BOOK 




Fis.216 Simple Pair of Gears 

In Fixed BearlnRB. 
Eq.l.N'a V. = E's V, x-i- 




Fis.2n Gears in Fixed Bearings. 

•with an Idler 
Eq.2.N'8 V. = B'B V. x-g- 



^^%,^ 




F '^R 

Fig. 2 18* .Sim pie Epicyclic Train 
Eq.3.N'8 V. =E'8 V.x (l-h-i-) 




/Fig. 219 Illustrating Rotation 

of N when it is Revolved 
about the Center of P. 



Fig.220 Second Stage in Deriving 
Equation 3:.Arm assumed to 
be fixed, F turned backward. 



Fig.221 Epicyclic Train 

w/ith an Idler 
E<^,4.i<'s V.=K'8 V. X (1-^) 




Fig. 222 Simple Epicyclic 

Train ^ith Internal Gear 
Eq.5.N'8T.= R'8 V. x 



Fig.223 Internal Gear Train with intei^ 
mediate Gear: the Arm Driving 
.Eq.e^N's V.= E'8 V. X (l^-X) 




Fig. 224 Same Train as Fig. 223 but 
•with the Internal Gear Driving 





Eq.7.N'8 V. = K'8 V. X (, 
+ 



x+-r 



■pig.225 Compounded Gears in 

Fixed Bearings 
Bq.8.N'8 V. =R'8 V. x^ 



Fig. 226 Compounded Gears In 
Fixed Bearings 
See Equation 8, Fig. 225 




Fig. 227 Compound Epicyclic Train 
Eq.9.N'8 V. =K'8 V. x (1— lllL) 



^^I^^ 





Fig. 228 Second Stage in Deriving Equation 9 
Arm assumed to be fixed, F turned 
backward 

Pig. 229 Compound Ei>icyclic Train -with 

One Internal Gear 
Ea.lO.N-8 V.= K'8 V. X (l-h-^) 

FIGS. 2l6 TO 229. 

EPICYCLIC GEAR TRAINS WITH CORRESPONDING VELOCITY RATIO FORMULAS. 



EPICYCLIC GEAR TRAINS 



255 



SIMPLE EPICYCLIC TRAIN 

Example III. 

Two gears, F and iV, are in mesh, the centers of which are on the arm i?, 
which is capable of revolving around the center of F. It is required to find 
the velocity ratio between R and N when R revolves around the fixed gear F; 




FIG.230. Compound Epicyclic Train with 
Two Internal Gears 
See EQ.9,sanie as Fig. 227 



ITOTATION 
K ==^ Denotes Driving Gear, or, in some 

cases, Arm. 
r = Number of Teeth in Driving Gear. 
N^ Denotes Driven Gear or Arm. 
n = Number of Teeth in Driven Gear. 
T, S and M denote Intermediate Gears. 
F = Denotes Fixed Gear. 
f = Number of teeth in it. 
V = Angular Velocity. 
y/////// Denotes part which is fixed.. 




FIG.231 An Epicylic Train Consisting of Two 
Central Gears, One Arm carrying Two 
Planetary Gears, and Two Internal 
Gears, One of which_is Eixed. 



FIG.2S2 Diagram of the Train of Fig 231 

Eqai.N'8 V.^K'sV.x /-r'n-rf \ 
^n(r+f/ 



FIGS. 230 TO 232. 
EPICYCLIC GEAR TRAINS WITH CORRESPONDING VELOCITY RATIO FORMULAS. 

Fig. 218 shows the arrangement. The gear N is subject to two motions due 
to the following two conditions: 

a. The fact of its being fixed to the arm R. 

b. The fact that it is in contact with the gear F. 

We will therefore in the first place suppose that they are not in gear, and 
that N cannot rotate on the arm R. Then if R makes one revolution around 
F it is obvious that N must also make one revolution around F, as in Fig. 219. 

.'. N^s velocity, due to condition a, = R's velocity, 

the direction being the same as R^s. 

Secondly, if instead of R making one revolution around F in a + direction, 
we cause F to make one in the opposite, that is, negative direction, we shall 



256 AMERICAN MACHINIST GEAR BOOK 

have exactly the same effect. Therefore place F and N in mesh, and fix the 
arm i?, as in Fig. 220. 

Then if F makes — i revolution, N will make H — revolutions. (Accord- 
ing to Equation i.) 

But - I of F = + I of i?. 

/ 
.'.I revolution oi R = - revolutions of N. 

n 

or, 

/ 
N^s velocity due to conditions b = ~ R's velocity. 

By addition we obtain the total impulses given in N, that is: 

N's velocity = R's velocity -{- ~ R's velocity 



R's velocity (1 + ~)' (3) 



EPICYCLIC TRAIN WITH IDLER 

Example IV. 

If an intermediate gear I be inserted between F and N , as in Fig. 221, we 
have a similar case to the above; but the intermediate gear has the effect of 
changing the direction of revolution of N (Equation 2), due to its contact 
with F through /. 

.'.N^s velocity = Rs velocity X (i — -) (4) 

It will be seen that if / = n,N will not have any motion of rotation at all; and 
it will have a positive one ii f<n and negative \i f>n. Thus by the ad- 
justment of / and n one can obtain great reduction in speed by means of few 
moving parts. 

• SIMPLE EPICYCLIC TRAIN WITH INTERNAL GEARS 

Example V. 

Instead of the driven gear N being external, it might have been internal, 
as shown in Fig. 222. The effect will be the same as inserting an intermediate 
gear in Example III, giving the same result as Case IV, namely: 

N's velocity = R's velocity X (i — -)• (5) 

In this case n>f. 

.'. The final direction is always +. 

INTERNAL GEAR EPICYCLIC WITH INTERMEDIATE GEAR 

Example VI. 

Fig. 223 shows a still further modification of this condition, I being an 
intermediate gear. The result is: 

N's velocity = R's velocity X (i-\--^y (6) 



EPICYCLIC GEAR TRAINS 257 

THE SAME TRAIN WITH THE INTERNAL GEAR DRIVING 

Example VII. 

With the above type, one often arranges the outer internal gear to be the 
driver, imparting motion to the arm carrying the intermediate gear (see 
Fig. 224). 

We have seen by equation 6 that: 

N^s velocity {driven) _ 1 
R's velocity (driver) 1 i + - I 

.*. N^s velocity = R^s velocity ^ I i — I 

= R's velocity X (^T^)* (?) 

The last two examples constitute what is known as the '' Sun and Planet" 
gear, which is largely used in many mechanisms. All the above examples 
show "simple" gearing, but they can be compounded with great advantage. 

COMPOUND GEARS IN FIXED BEARINGS 

Example VIII. 

Gears compounded together are shown in Figs. 225 and 226, 226 being a 
diagram of 225. One repeats the well-known rule that: 

Velocity of driven gear _ Product of number of teeth of driver gears 
Velocity of driver gear Product of number of teeth of driven gears 

r ^^ m 

or, N^s velocity = R^s velocity X ^ (8) 

s X n 

The direction is the same as iV'5, namely, +. 

COMPOUND EPICYCLIC TRAIN, WITHOUT INTERNAL GEAR 

Example IX. 

We will now arrange to fix one of the gears F, and by means of the arm R 
revolve the others around it, thereby causing N to revolve as shown in Figs. 
227 and 228. As before, we will assume the gears M and 5 to be out of mesh, 
so that when the arm R, carrying with it the gear TV, makes one revolution 
around F, N must also make one revolution relatively to F. Also when they 
are in mesh, the arm R being fixed and F makes one revolution in a negative 

direction (see Fig. 228), N will make — — revolutions. (Equation 8.) 

Now the total motion imparted to N must be the sum of these two, 
namely: 

I revolution of R = 1 — - — revolutions of N, 

■' sn •' 



or, 



(f X m\ 
I — '^—— — I • (9) 

5 /\ n / 



258 AMERICAN MACHINIST GEAR BOOK 



COMPOUND EPICYCLIC TRAIN WITH ONE INTERNAL GEAR 

Example X. 

Fig. 229 shows a slight modification of the last case, iS being an internal 
instead of an external gear. Obviously the only difference will be in the direc- 
tion of N^s motion, that is: 

I + - — I- (10) 

COMPOUND EPICYCLIC TRAIN WITH TWO INTERNAL GEARS 

Example XI. 

A further modification, however, is one in which both F and N are internal 
gears (Fig. 230), the effect of such being a change of sign in the equation. 

(fm \ 
I — - — I . (9) 

The type shown in Figs. 227 and 230 is, perhaps, one of the best methods 
of obtaining a good reduction of speed in an easy and cheap manner. 

There are several combinations of the examples shown, but as they are all 
somewhat similar we will take another typical case as a guide for future 
calculations. 

AN EPICYCLIC TRAIN CONSISTING OF TWO CENTRAL GEARS, ONE ARM 

CARRYING TWO PLANETARY GEARS, AND TWO INTERNAL GEARS, 

ONE or WHICH IS FIXED 

Example XII. 

The writer has successfully used the arrangement shown in Figs. 231 and 
232, in which R and R' are two spur gears mounted on one shaft; / and /' 
are two ''planet" pinions, while F and A^ are two internal gears, the former 
being fixed. R and R' are made to revolve, which has the effect of giving N 
a very slow speed. 

A SCHEME FOR FINDING THE VELOCITY RATIO 

As this is somewhat complicated, we will work it out in stages: 

1. Obtain the revolutions of the arm A when R' makes one revolution, F, 
of course, being fixed. 

2. Obtain N's revolutions when the arm A is fixed and R makes one 
revolution. 

3. Assume R fixed, and that the arm makes one revolution; obtain, then, 
iV'5 revolutions. 

4. Then if A^ makes so many revolutions to one of the arm, as given by 
stage 3, we can by proportion obtain how many will be caused by the amount 
given by Stage i. 



EPICYCLIC GEAR TRAINS 259 

5. Add the results of 2 and 4 together, and obtain the motion given to N 
by one revolution of R, which is the desired result. 

THE SCHEME WORKED OUT 

Working the above out we obtain: 

I. When F is fixed and R' makes one revolution, the arm A must make + 



r' 



(According to Equation 7.) 



y 

2. i? makes one revolution, arm A being fixed; then iV must make — - 

revolutions. (According to Equation 2. Negative sign used because of the 
internal gear.) 

3. When R is fixed and arm A makes one revolution, iV will make 

+ \r + -) revolutions. (According to Equation 6.) 

r 

4. With one revolution of arm, N makes i H — revolutions, from Stage 3; 

r' 
.'. with -T", — r revolutions of the arm, as derived in Staee i, N will make 

r -\- J 



(^ + rJ X (7T7) 



5. The aggregate is the sum of the effects derived in Stages 4 and 2, 
namely, to one of i?, N makes: 

\ n/ \r +// \ n/ n{r (Ef) n 
rr' + r'n — rr' — ^f _ ^'^ — ^f 

The final direction of revolution of iV will depend upon the relation which 
/n bears to rf; if the former be greater, then the direction will be positive 
(+), and vice versa. The formula for this combination is then: 

(T it — ff \ 
( ^J- A )• ^^^) 

SOME NUMERICAL EXAMPLES IN EPICYCLIC GEARING 

In order to illustrate the above examples we will take one or two cases. 
If in Example and Fig. 218,/ = 30, /e = 25, then to one revolution of i?, 

N will make (i+j = 1 + ^^^5 = 2^:5 revolutions. 

It will be obvious that with f = n, N would revolve at twice the speed oiR. 
In the type shown in Fig. 7, / = 60, n = 65; 
then 

I 

Velocity of N ^ ^ ~ n ^ i - ^% 5 _ 5 ^ ^1/ 
Velocity of R i i "^^ ^^^° 



26o 



AMERICAN MACHINIST GEAR BOOK 



The arrangement of Fig. 227 is much used. Let n = 60, f = 61, s = 40, 

m = 41. 

fm 



sn 






Then the velocity ratio between N and i? is i 

61 X 41 2,501 . . ,. - 

- = say 1:24, m a mmus direction. 



= I 



40 X 60 2,400 

Illustrating Example XII, Fig. 231, let r = 90,/ = 91,/ = 120, w = 121. 
Velocity of N r'n — rf 91 X 121 — 90 X 120 



Velocity of R n{/ -\- f) 
11,011 — 10,800 
121 X 211 



121(91 + 120) 
211 _ I 
121 X 211 121 



Arm 



DIRECTION OF ROTATION OF GEARS 

The following, in reference to epicyclic gear calculations, is by Oscar J 
Beale, American Machinist, July 9, 1908: 

"A very valuable article relative to epicyclic gears is by Prof. A. T. Woods 
in the American Machinist for February 14, 1889. This article is well- 
nigh perfect. It is so clear and com- 
prehensive that it was of great help to 
me. I have read a number of later 
articles, and I have always gone back 
to this in order to clear up the subject. 
I think that many of your readers 
would like to see it reprinted. 

''About the best way to determine 
the direction of rotation of epicyclic 
gears is by careful inspection of the posi- 
tion of the members; then, if you make a 
correct statement of the effect of each 
member. Professor Woods' methods 
will bring the answer right. One must 
be sure to give the result the proper 
sign; that is, one mu&t be able to add 
and to subtract algebraically. 
"I have sometimes used a sort of mental key in cases like this sketch. 
Fig. 233. If the pitch circle of L is smaller than the pitch circle of F, the 
rotation of L will be opposite that of the arm. If L is greater than F, the 
rotation of L will be the same as that of the arm. 

"This 'mental key' may help some; but after all, it is usually better to 
reason mathematically as in Professor Wood's article." 

We reprint herewith Professor Woods' article to which Mr. Beale refers 
because of its value in determining the direction of rotation of epicyclic trains. 



Fixe(i 



FIG. 233. 



F L 



DIAGRAM OF AN EPICYCLIC 
TRAIN 



EPICYCLIC GEAR TRAINS 



261 



EPICYCLIC TRAINS* 

"An epicyclic train consists of a number of gear wheels, or pulleys, and 
belts, some of which are carried upon a revolving arm. For example, in Fig. 
236 the wheel F is fastened to the shaft B, about which arm A turns. This 
arm carries the axes of C and L, C being an idle wheel gearing with F and L. 
The motion of the wheel L is thus composed of three motions : (i) That which it 
has by reason of its revolution about 5 as a center, (2) that due to the revo- 
lution of the arm A about F, and (3) that due to its connection with F by 
means of the wheel C. We will consider the effect of these motions sepa- 





Fig.236 



Fig. 234 





Fig. 237 



DIAGRAMS OF EPICYCLIC GEARS 



rately, and will begin with the simplest possible arrangement. In Fig. 234 
let A be an arm which revolves about a center B, and carries a wheel L, which 
we will suppose to be fastened to it. If the arm be turned through one revo- 
lution, the wheel L will in effect revolve once about its own center. This will 
be clear by an examination of the successive positions shown in dotted lines, 
the revolution of the arm being in the direction of the arrow. For example, 
follow the motion of a point such as P; at i it is to the right of the center, at 2 
below it, at 3 to the left, and at 4 about the center, finally returning to its first 
position on the right. We thus see that L has practically made one revolution 

*Prof. A. T. Woods. 



262 AMERICAN MACHINIST GEAR BOOK 

about its own center, just as it would have if it had been fixed at B concentric 
with the arm. If L has not revolved by reason of the revolution of A^ the 
point P would have remained horizontally to the right of the center during the 
revolution of A. This is, of course, the same motion as that of a crank-pin 
and crank, and will be still more clear if we remember that, if the pin did not 
in effect revolve about its own center, it would not turn in the brasses, and 
they could be dispensed with. 

Now, considering the second motion of L, that due to the revolution of the 
arm about F, refer to Fig. 235, and let the wheel F be fixed, or a dead wheel, 
concentric with the arm A. Let F and L have the same number of teeth. 
Then while the arm revolves once in the direction indicated, L will revolve in 
the same direction, the result being the same as if the arm had remained fixed, 
and F had been revolved once in the opposite direction. The final motion of 
L, while the arm revolves once, is therefore one revolution, as in Fig. 234, 
supposing F to be removed, and one revolution, as in Fig. 235, supposing F 
to be in place and fixed; or, while the arm revolves once, L revolves twice in 
the same direction. 

Now, instead of F being fixed, let it revolve once in the direction of the 
dotted arrow. The effect of this will be to give one additional revolution to L, 
resulting in three revolutions of L to one of the arm. In order, then, to get at 
the resultant motion of the last wheel in epicyclic trains, we must consider the 
three independent motions separately : First, suppose the first wheel F, which 
is concentric with the arm, to be removed; second, suppose the first wheel to 
be in place and fixed; and third, suppose the arm to be fixed and the first wheel 
to revolve as intended. The final motion of L is the sum of these three 
motions. 

For the sake of brevity we will designate revolution in the direction of the 
hands of a watch ahead, or + ; and that in the opposite direction backward, 
or — . Thus expressed, the revolutions of L in Fig. 235, as we have just dis- 
cussed it, will be: 

(a) + I due to the revolution of the arm, 
+ I due to the revolution about F", 
+ I due to the revolution of F. 

+ 3 revolution of L to one of the arm. 

As a further illustration assume that in Fig. 235, F has 40 teeth and L 30, 
then if F is a fixed wheel, L will revolve: 

(b) + I due to the revolution of the arm, 
+ '^/^o due to the revolution about F, 

o due to the revolution of F, 



+ J^ revolutions of L to one of the arm. 

If F makes one revolution backward while the arm makes one ahead, we 
will have for L : 



EPICYCLIC GEAR TRAINS 263 



(c) + I due to the revolution of the arm, 
-i~ ^^'30 due to the revolution about F, 
+ ^/^o due to the revolution of F, 



+ 13^^ revolutions of L to one of the arm. 
If F makes one revolution ahead, or in the same direction as the arm, the 
result is to balance the effect of the revolution about F, and we have for L: 
(d) + I due to the revolution of the arm, 

+ ^%o due to the revolution about F, 

— ^%o due to the revolution of F, 



+ I revolution of L to one of the arm, or the same as in Fig. 234. 

Similarly, if we take the conditions the same as (c) and let F 

have 30 teeth and L 40, we will have + 23^^ revolutions of L 

to one of the arm. 

We will now consider the effect of introducing an idle wheel, as shown in 

Fig. 236. In the first place, let L equal F, and let F be a fixed wheel. The 

revolutions of L will be: 

(e) + I due to the revolution of the arm, 

— I due to the revolution about F, 
o due to the revolution of F, 

o revolution of L, or, in other words: A point P, which is, say, ver- 
tically over the center of L, will remain so throughout the revo- 
lution of the arm. If we assume the same condition as (a), the 
resulting revolution of L will be: 

(f) + I due to the revolution of the arm, 

— I due to the revolution about F, 

— I due to the revolution of F, 



— I revolution of L to one of the arm. 

If we let F have 40 teeth and L 30, and let F be a dead wheel, we will have 
forZ: 

(g) + I due to the revolution of the arm, 
~ ^%0 due to the revolution about F, 
o due to the revolution of F, 



— y^ revolution of L to one of the arm. 

Or, reversing the position of the wheels, making F = 30 and L = 40, the 
revolutions of L will be: 

(h) -h I due to the revolution of the arm. 



_ 40Z 



^0 due to the revolution about F, 



o due to the revolution of F, 



+ y^i revolution of L to one of the arm. 



264 



AMERICAN MACHINIST GEAR BOOK 



Tt 



•B 



Fig. 238 



g 



-H 



-M 



The arrangement shown at (e) is used in one form of rope-making machin- 
ery. The "arm" A, Fig. 236, is then the revolving frame which carries the 
bobbins on which the strands or wire have been wound. B is the center of 
this frame, and on it the wheel F is fixed. A small yoke or frame, which 
carries a bobbin, is fixed on the axis of L, there being as many of these wheels 
and bobbins as there are to be strands in the rope. Then if F and L have the 
same number of teeth as at (e), the axes of the bobbins always point in one 
direction, and the rope is laid up without twisting the separate strands. If L 
has a few less teeth than F, the strands will be given a slight twist, making the 
rope harder. 

Arrangements such as Figs. 235 and 236 are applicable to boring bars 

having sliding head. In such cases B would be the dead center on which the 

bar turns and on which the wheel F is fast- 
is 

ened, being, therefore, a dead wheel. The 
wheel L is fastened to the end of the feed- 
screw in the bar, as shown at S, in Fig. 237, 
which represents the end view of the bar. 
While the arrangement is an epicyclic 
train, such as we have discussed, the 
explanation of it is extremely simple, be- 
cause the motion to be determined is that 
of the screw S with regard to the bar, not 
with regard to the lathe, or any stationary 
object. 

As F is fixed, the effect on L of one 
revolution of the bar is the same as if the 
bar remained stationary and F revolved 
once. Thus, if F has 20 teeth and L 40, the 
screw S will make, in the bar, 2%o = M of a revolution, while the bar 
revolves once. And if the screw has four threads to the inch, the feed will 
be H X 3^^ = ^i inch. The effect of the idle wheel (shown in Fig. 237) is 
simply to change the direction of the feed. 

Another form of epicyclic train is that shown in Fig. 238 in which the last 
wheel is concentric with the arm and first wheel. This does not change the 
resultant motion of L in any way, but only makes a more convenient form for 
transmitting motion from L to other parts of the machine. If F and L have 
40 and 30 teeth, while the wheels C and D are equal, we will have the same 
motion as at (g) and (h), supposing F to be a dead wheel. A recent and novel 
application of this form of train is to be found in the Waterbury watch, the 
principle of which is shown in Fig. 239. In this figure ah is the face of the 
watch and c d the frame which carries the principal train of wheels, that from 
C to the balance wheel g. This frame turns about the center shown below it 
and the bearing in the face. It is driven by the spring e and carries the 
minute hand M, and hence revolves once each hour. Between the frame and 



ZJ" 

□ 

Fig. 239 
Waterbury Watch Movement. 



EPICYCLIC GEAR TRAINS 265 

the face is a pinion C, having 8 teeth, which is connected to the balance 
wheel g by the train of wheels shown, and itself gears with two wheelsFand L, 
having, respectively, 44 and 48 teeth. F is fastened to the face, and so is a 
dead wheel, and L is fastened to the hour hand H by the tube, as shown. It 
remains to show that the hour hand will revolve once in 1 2 hours, as required 
by means of this connection. We have here an epicyclic train FCL in which 
the first wheel is fixed. The revolutions of L during one revolution of the arm, 
as we have called it, or the frame cd are, therefore: 

+ I due to the revolution of the arm, 
— ^^8 due to the revolution about F, 
o due to the revolution of F, 



+ Ms = + M2 revolution during one revolution of the arm or minute 
hand, which is, of course, as it should be. The remainder of the train of 
wheels in this watch do not differ in principle from that ordinarily employed, 
a peculiarity being, however, that the entire "works," held in the frame cd, 
revolve within the case every hour. 

Another peculiar adaptation of epicyclic trains is for the production of 
very slow velocities, using a small number of wheels. For example, in Fig. 
238, let the numbers of teeth on the several wheels beF = 19, C = 20, Z) = 21, 
and L = 20, and let F be a dead wheel. Working out this train as we have 
the others, it will be found that, while the arm A makes one revolution, the 
last wheel / will make but Moo o^ 9- revolution. In the same way, if we take 
the number of teeth in order as above, as 27, 40, 37 and 25, the last wheel will 
make but one revolution, while the arm makes 1,000. 

We have chosen examples in which the first wheel is the dead wheel, as 
these are the simplest and most common. By adjusting the speed of the first 
wheel, however, it becomes possible to transmit velocities by means of epi- 
cyclic trains, which would be practically impossible by ordinary means. As an 
illustration, suppose it is required to have one shaft make 641 revolutions to 
one of another. As 641 is a prime number, this ratio could not be trans- 
mitted exactly by ordinary gearing on account of the large number of teeth 
required for a single wheel; but by means of an epicyclic train it can be readily 
accomplished. Of course, the necessity for such ratios as this rarely occurs 
in machinery. 

A method of solving problems involving epicyclic trains, which will be 
more convenient for many than that which we have followed, is by means of a 
general formula. Let v = the value of the train of gears, or the product of 
the number of teeth on the drivers divided by the products of the numbers of 
teeth on the followers, which would be, in Fig. 238, 

FXD 
CXL' 

In case of pulleys, v = the product of the diameters of the drivers divided by 
the product of the diameters of the followers. Let /, / and a represent the 



266 



AMERICAN MACHINIST GEAR BOOK 



number of revolutions of the first wheel, last wheel, and arm, respectively, in 
the same time. 
Then, 

I — a 

V = . 

/- « 
If one direction is represented as +, the other will be represented as — . 
If the last wheel is to revolve in the same direction as the first, supposing the 
arm to be fixed, ^^ is +, and if the opposite direction, it is — . For example, 
take the data as at (b) ; then. 

V = ^%o = ' whence I = J"^ a. 

o — a 

Again, let it be required that L shall make one revolution to i,oooof ^ 
(Fig. 238), 



I — 1,000 , QQQ 

V = = + 



o — 1,000 



2 7 X 37 _ F X D 

1,000 40 X 25 C X L 



TABLE OF PROPORTIONS OF DIFFERENTIAL BACK-GEARS 

The following table, originally pubHshed in American Machinist by 
Ernest J. Lees, gives data for ready reference. For a back-gear on drill 
presses and other light machinery, the differential back-gear as originally 



SIZE NO. 


I 


2 
8 


3 


4 


5 


6 


7 


8 


9 


10 


II 


12 


Diameter of Pulley 


8 


10 


10 


12 


12 


15 


15 


15 


15 


18 


18 


Face of Pulley 


3 


3 


s'A 


3V2 


43^ 


4M 


Sli 


s'A 


6^ 


6M 


7H 


7y2 


Width of Belt 


2y2 


2H 


3 


3 


4 


4 


5 


5 


6 


6 


7 


7 


Approximate H. P. at 
300 r. p. m. 


2H 


2i.^ 


4H 


4'i 


7H 


7'i 


II 


II 


13 


13 


18 


18 


Shaft Diameter D 


i3^ 


1V2 


IM 


iH 


zVs 


^Vs 


^H 


i^ 


2 


2 


3 


3 


Pitch Diameter pinion 
A 


2H 


4H 


2) 


5l 


2*r 


s) 


2) 


6 


3 


9 


5 


13 


Number of teeth in A 


18 


36 


18 


36 


I& 


36 


18 


42 


18 


54 


15 


39 


Pitch Diameter Inter- 
nal Gear B 


9 


9 


IO» 


10? 


107 


10" 


I2f 


I2f 


18 


18 


25 


25 


Number of teeth in B 


72 


72 


72 


72 


72 


72 


90 


90 


108 


108 


75 


75 


Pitch Diameter Idler C 


3% 


2M 


31 


2) 


3? 


2| 


s) 


3? 


7>^ 


4M 


10 


6 


Number of teeth in C 


27 


18 


27 


18 


27 


18 


36 


24 


45 


27 


30 


18 


Number of Idlers 


3 


3 


3 


3 


3 


3 


3 


3 


3 


3 


3 


3 


Diameter Pitch of 
Gears 


8 


8 


7 


7 


7 


7 


7 


7 


6 


6 


3 


3 


Face of Gears 


iH 


iM 


iH 


I'A 


i^ 


iVs 


iH 


iM 


2 


2 


3 


3 


Ratio 


5 
to 


3 

to 


5 
to 


3 
to 


5 
to 


3 
to 


6 
to 


3-142 
to 


7 
to 


3 
to 


6 
to 


2.923 
to 




I 


I 


I 


I 


I 


1 


I 


I 


I 


I 


I 


I 



TABLE 24. PROPORTIONS OF DIFFERENTIAL BACK-GEARS. 



EPICYCLIC GEAR TRAINS 



267 



designed. For a heavy drive and continuous service there is a better method 
of arrangement. This consists of using three idlers in place of one, these being 
equally spaced in order to retain the balance of the whole when locked up and 
driving direct. It will be readily seen that this arrangement calls for the 
following conditions in the gearing: That the number of teeth in pinion, idler, 




FIG. 240. DIFFERENTIAL BACK-GEARING. 



Stafionary 



and internal gear must each be divisible by three, at the same time having 
correct diameters and pitch. 

A table is given herewith showing drives from 23^^ to 18 horse-power in 12 
sizes. This is arranged so that there are only 6 different diameters of internal 
gears, and by using different pinion and idlers one size can be used for two 
different ratios. 

Fig. 240 shows the principle on which these back-gears are operated. The 
locking device should not be copied, however, as it would be rather incon- 
venient to use, and as there are various methods of operating clutches the 
writer has not gone into detail on this point, but gives the general dimensions, 
leaving the rest to be worked out by the designer to suit the requirements 
and conditions of the machine on which it is to be used. 



SECTION XIII 

Friction Gears* 

A friction drive, as the term is here employed, consists of a fibrous or some- 
what yielding driving wheel working in rolling contact with a metallic driven 
wheel. Such a drive may consist of a pair of plain cylindered wheels mounted 
upon parallel shafts, or a pair of beveled wheels, or of any other arrangement 
which will serve in the transmission of motion by rolling contact. The use of 
such drives has steadily increased in recent years, with the result that the so- 
called paper wheels have been improved in quality, and a considerable num- 
ber of new materials have been proposed for use in the construction of fibrous 
wheels. 

THE WHEELS TESTED 

Choosing materials which have been used for such purposes, driving 
wheels of each of the following materials have been tested: straw fiber, straw 
fiber with belt dressing, leather fiber, leather, leather faced iron, sulphite 
fiber, tarred fiber. 

The straw fiber wheels are worked out of the blocks which are built up usu- 
ally of square sheets of straw board laid one upon another with a suitable 
cementing material between them and compacted under heavy hydraulic 
pressure. In the finished wheel the sheets appear as disks, the edges of which 
form the face of the wheel. The material works well under a tool, but it is 
harder and heavier than most woods and takes a good superficial polish. The 
wheel tested was taken from stock. 

The wheel of straw fiber with belt dressing was similar to that of straw 
fiber, except that the individual sheets of straw board from which it was made 
had been treated, prior to their being converted into a block, with a "belt 
dressing" the composition of which is unknown to the writer. 

The leather fiber wheel was made up of cemented layers of board, as were 
those already described; but in this case the board, instead of being of straw 
fiber, was composed of ground sole leather cuttings, imported flax and a small 
percentage of wood pulp. The material is very dense and heavy. 

The leather wheel was composed of layers or disks of sole leather. 

The leather faced iron wheel consisted of an iron wheel having a leather 
strip cemented to its face. After less than 300 revolutions the bond holding 
the leather face failed and the leather separated itself from the metal of the 

* Abstract of paper presented to the American Society of Mechanical Engineers, 
December, 1907, by W. M. Goss, Professor, University of Illinois. 

268 



FRICTION GEARS 



269 




wheel. This wheel proved entirely incapable of transmitting power and no 
tests* of it are recorded. 

The wheel of sulphite fiber was made up of sheets of board composed of 
wood pulp. The sulphite board is said to have been made on a steam-drying 
continuous process machine in the same way as is the straw board. 

The tarred fiber wheel was made up of board composed principally of 
tarred rope stock, imported French flax, and a small percentage of ground 
sole leather cuttings. 

Each of the fibrous driving wheels was tested in combination with driven 
wheels of the following materials: Iron, aluminum, type metal. All wheels 
tested, both driving and driven, were 16 
inches in diameter. The face of all driving 
wheels was 1% inches while that of all 
driven wheels was J^ inch. 

The purpose of the experiments was to 
secure information which would permit 
rules to be formulated defining the power 
which may be transmitted by the various 
combinations of fibrous and metallic wheels 
already described. To accomplish this it 
was necessary to determine for each com- 
bination of driving and driven wheel the 
coefficient of friction under various condi- 
tions of operation ; also the maximum pressures of contact which can be with- 
stood by each of the fibrous wheels. 

The testing machine used is shown diagrammatically by Fig. 241. The 
principles involved will be made clear by assigning the functions of the actual 
machine to the several parts of this figure. The shaft A runs in fixed bearings 
and carries the fibrous friction wheel. This wheel is the driver. Its shaft A 
carries, besides the friction wheel, two belt pulleys, one on either side, which, 
from any convenient source of power, serve to give motion to the driver. The 
shaft B carries the driven wheel, which in every case was of metal. The bear- 
ings of this shaft are capable of receiving motion in a horizontal direction and 
by means of suitable mechanism connected therewith, the metal driven wheel 
may be made to press against the fibrous driver with any force desired. The 
pressure transmitted from B to A is hereinafter referred to as the "pressure of 
contact" and is frequently represented by the symbol F. The tangential 
forces which are transmitted from the driver to the driven wheel are received, 
absorbed and measured by a friction brake upon the shaft B. In action, 
therefore, the driven wheel always works against a resistance, which resist- 
ance may be modified to any desired degree by varying the load upon the 
brake. The theory of the machine assumes that the energy absorbed by the 
brake equals that transmitted from the driver to the driven wheel at the con- 
tact point C, Accepting this assumption, the forces developed at the peri- 



w////wwy//// ■ 



FIG. 241. DIAGRAM OF TESTING MA- 
CHINE FOR FRICTION WHEELS. 



270 



AMERICAN MACHINIST GEAR BOOK 



phery of the brake wheel may readily be reduced to equivalent forces acting 
at the circumference of the driven wheel. The force, which is directly trans- 
mitted from the driver to the driven wheel, is hereinafter designated by the 
symbol F. It will be apparent from this description that the functions of 
the apparatus employed are such as will permit a study of the relationship 
existing between the contact pressure P and the resulting transmitted force 




E 



Partial Elevation 
FIG. 242. THE TESTING MACHINE FOR FRICTION WHEELS. 



F, which relation is most conveniently expressed as the coefficient of friction. 
It is, 



It is obvious, in comparing the work of two friction wheels, that the one 
which develops the highest coefficient of friction, other things being equal, 
can be depended upon to transmit the greatest amount of power. 

The actual machine as used in the experiments is shown by Fig. 242. Its 
construction satisfies all conditions which have been defined except that shaft 
JB, Fig. 241, does not run in bearings which are absolutely frictionless, as is 
required by a rigid adherence to the theoretical analysis already given. 
These bearings, however, are of the "standard roller bearing" type and of 
ample size, and it is believed that the friction actually developed by them is so 



FRICTION GEARS 271 

small compared with the energy transmitted between the wheels that it may 
be neglected. 

The bearings of the fixed shaft A are secured to the frame of the machine. 
The bearings of the axle B are free to move horizontally in guides to which 
they are well fitted. Those bearings are connected by links to the short 
arm of a bell crank lever, the arm of which projects beyond the frame of the 
machine at the right-hand end and carries the scale pan and weights E. The 
effect of the weights is to bring the driven wheel in contact with the driver 
under a predetermined pressure, the proportions of the bell crank lever being 
such as to make this pressure in pounds equal, 

P = 10 W + 73, 

where W is the weight on the scale pan E. 

The fulcrum of the bell crank lever is supported by a block G which may 
be adjusted horizontally by the hand wheel H at the rear of the machine, so 
that whatever may be the diameter of the driven wheel, the long arm of the 
bell crank may be brought to a horizontal position. The constants employed 
in calculating the coefficient of friction from observed data are as follows: 

Diameter of friction wheels (inches) 16 

Effective diameter of brake (inches) 18.35 

Ratio of diameter of friction wheel to that of brake wheel 1.145 
Effective load on brake F^ 

F' 

Coefficient of friction i-i45 ^ 

The slippage between the friction wheels was determined from the 
readings taken from the counters connected to each one of the shafts. 

THE TESTS 

In proceeding with a test, load was applied to the scale pan E, Fig. 242, to 
give the desired pressure of contact, after which the hand wheel H at the back 
of the machine was employed to bring the bell crank to its normal position. 
This accomplished, with the driving wheel in motion, the driven wheel would 
roll with it under the desired pressure of contact. A light load was next 
placed upon the brake to introduce some resistance to the motion of the 
driven shaft, and conditions thus obtained were continued constant for a 
considerable period. Readings were taken simultaneously from the counters 
and time noted. After a considerable interval the counters were again read, 
time again noted, and the test assumed to have ended. From the readings of 
the counters and from the known diameters of the wheels in contact, the 
p>ercentage of slip attending the action of the friction wheels was calculated. 
Three facts were thus made of record, namely: (a) The pressure of contact. 



272 



AMERICAN MACHINIST GEAR BOOK 



(b) the coefficient of friction developed, and (c) the percentage of slip resulting 
from the development of said coefficient of friction. 

This record having been completed, the load upon the brake was increased 
and observations repeated, giving for the same pressure of contact a new 



































5 


















































r 


^ 







_c 




— 


0.4 
















U;' 
























^y^ 


^0' 




















0.3 








/ 




























/ 


r 
























0.2 






/ 






























/ 


























0.1 












St 


■av 


Fi 


aer 


IrJ 


m 


















1 


50 Pon 


Ids Pr 


ess 


ire 







































































0.5 

















































































-D- 


— 


— 


p-S 


















,/^ 




















0-1 










/ 






























f 






















'0 

e 0.2 




































/ 


























0.1 






6 






Straw 


Fi^r 


Trqn 


















^00 ^on 


ids} Pijessure 
























1 















12 3 12 3 

Slip, Percent Slip, Percent 

FIG. 243. FIG. 244. 

CURVES FOR STRAW FIBER AND IRON, TYPICAL FOR ALL CURVES PLOTTED FROM THE 

FRICTION TESTS 



coefficient of friction and a higher percentage of slip. This process was 
continued until the slippage became excessive and in consequence thereof the 
rotation of the driver ceased. By this process a series of tests was developed 
disclosing the relation between slip and coefficient of friction for the pressure 

in question. Such a series having been 
completed, the load upon the weight 
holder E was changed, giving a new 
pressure of contact, and the whole pro- 
cess repeated. As the work proceeded, 
curves showing the relation of coefficient 
of friction and slip for pressures per inch 
width of face in contact of 150 pounds 
and 400 pounds, respectively, were secur- 
ed. The curves shown by Figs. 243 and 
244 for the straw fiber driving wheel in 
contact with the iron driven wheel are 
typical in their general form of those obtained from all combinations of 
wheels, but the curves of no two combinations were alike in their numerical 
values. 

Having completed this series of tests at constant pressure, a series was 
next run for which the coefficient of shp was maintained constant at 2 per 
cent., and the pressure of contact varied from values which were low to those 
which are judged to be near the maximum for service conditions, with the 
results, which in all cases were similar in character with those given for the 
straw fiber and iron wheels, as set forth by Fig. 245. The numerical values of 



































C.5 






























































( 


Y- 


U-, 




^ — i 






















^0-^ 






















1 










«M 05 

































































■§0.2 






























































00.1 










St 


'a\ 


I' -Fiber [ 


rot 


. 


















2 plercjent 


Slip 
























_ 


_ 


_ 















150 



200 250 300 350 
Contact Pressure 



400 



FIG. 245. CURVE FOR STRAW FIBER AND 
IRON WITH CONSTANT SLIP. 



FRICTION GEARS 273 

points for other combinations were not the same as those shown by Fig. 245, 
but in the case of most of the combinations the coefficient of friction at 
constant sHp gradually diminishes as the pressure of contact is increased. 
As the series of tests involving each combination of wheels proceeded, 
the increase in pressure of contact was discontinued when the markings made 
upon the driving wheel by the metallic follower became so distinct as to 
suggest that a safe limit had been reached; but when all other data had been 
secured, tests were run for the purpose of determining the ultimate resistance 
of the fibrous wheel to crushing. The details of these will be described later. 

COEFFICIENT OF FRICTION DEVELOPED BY THE SEVERAL COMBINATIONS OF 

WHEELS — STRAW FIBER AND IRON 

The results of experiments involving a straw fiber driver and an iron driven 
wheel a.re shown graphically in Figs. 243, 244, and 245. Figs. 243 and 244 
illustrate the relation between slip and coefficient of friction when the two 
wheels are working together under pressures per inch width of 150 and 400 
pounds, respectively. 

The figures show that although the values of the coefficient of friction are 
slightly lower than corresponding ones for 150 pounds pressure, the curves 
are sufficiently similar to establish the fact that the law governing change in 
coefficient friction with slip is independent of the pressure of contact. When 
the slippage is 2 per cent, the coefficient of friction is 0.425 for a contact 
pressure of 400 pounds. That the coefficients of friction for all pressures 
between the limits of 1 50 pounds and 400 pounds are practically constant is 
well shown by the diagram Fig. 245. The pressure of 400 pounds is the 
maximum at which tests of this combination of wheels were run, though 
straw fiber was successfully worked up to a pressure of 750 pounds. 

STRAW FIBER AND ALUMINUM 

By curves plotted from values for a straw fiber driver and aluminum 
driven wheel, it can be shown that when the working pressure is 150 pounds 
per inch width and the slippage is 2 per cent, the coefficient of friction is 
0.455; ^^so, that for all pressures ranging from 100 to 400 pounds, the coeffi- 
cient of friction is practically constant when the rate slip is constant. The 
maximum pressure at which tests involving this combination of wheels were 
run was 400 pounds per inch width. 

STRAW FIBER AND TYPE METAL 

By curves plotted from values for a straw fiber driver and a type metal 
driven wheel it can be shown that when the two wheels are operated under a 
pressure of contact of 150 pounds per inch width and when the slip is 2 per 
cent, the coefficient of friction is 0.310; also, that for all pressures of contact 

18 



2 74 AMERICAN MACHINIST GEAR BOOK 

ranging from loo to 400 pounds, the coefficient of friction is practically 
constant when the sHp is constant. 

STRAW FIBER WITH BELT DRESSING AND IRON 

Curves plotted from values for a straw fiber driver treated with belt 
dressing, and an iron driven wheel show that when the two wheels are worked 
together under a pressure of 1 50 pounds per inch width and when the slip is 
2 per cent, the coefficient of friction is 0.12; also, that for all pressures up to 
400 pounds per inch width, the coefficient of friction remains constant. The 
greatest pressure at which tests of this combination of wheels were run was 
500 pounds per inch width. 

LEATHER FIBER AND IRON 

Curves plotted from the results of tests involving a leather fiber driver and 
an iron driven wheel show that when the two wheels are worked together 
under pressure of 150 pounds per inch in width and when slip is 2 per cent, the 
coefficient of friction is 0.515. When the contact pressure is 300 pounds per 
inch width, the coefficient of friction is 0.510. The greatest pressure at 
which tests of this combination of wheels were run was 350 pounds per inch 
width, although leather fiber was successfully worked up to a pressure of 
1,200 pounds per inch width. 

LEATHER FIBER AND ALUMINUM 

Curves plotted from the results of experiments involving a leather fiber 
driver and an aluminum driven wheel show that under a contact pressure of 
150 pounds per inch width and a slip of 2 per cent, the coefficient of friction 
is 0.495. 

This value remains practically constant under all pressures. The 
maximum pressure used in tests of this combination of wheels was 400 
pounds. 

LEATHER FIBER AND TYPE METAL 

Curves plotted from the results of experience involving a leather fiber 
driver and a type metal driven wheel show that when the wheels are operated 
under a contact pressure of 150 pounds per inch width and when the slip is 
2 per cent, the coefficient of friction remains constant for all pressures up to 
400 pounds per inch width. 

TARRED FIBER AND IRON 

Curves plotted from the results of the experiments involving a tarred fiber 
driver and an iron driven wheel show that the change in the value of the 
coefficient of friction with change of slip is practically independent of the 



FRICTION GEARS 275 

pressure of contact. When the sHp is 2 per cent., the coefficient of friction is 
0.220 for a pressure of contact of 150 pounds and 0.250 for a pressure of 
contact of 400 pounds per inch width. 

Tests of this combination were made also under different speeds when the 
wheels were working together under a pressure of contact of 250 pounds per 
inch width and when the slip was 2 per cent., with the result that the coeffi- 
cient of friction was found to remain nearly constant for speeds of 450 and 
3,350 feet per minute, respectively. The greatest pressure at which tests of 
this combination of wheels were run was 400 pounds per inch width, although 
tarred fiber was successfully worked up to a pressure of 1,200 pounds per inch 
width. 

TARRED FIBER AND ALUMINUM 

Curves plotted from the results of experiments involving a tarred fiber 
driver and an aluminum driven wheel show that when the slip is 2 per cent, 
and the pressure of contact 150 pounds per inch width, the coefficient of 
friction is 0.305; also, that for a pressure of 400 pounds per inch width, the 
coefficient of friction is 0.295. The greatest pressure at which tests of this 
combination were run was 400 pounds per inch width. 

TARRED FIBER AND TYPE METAL 

Curves plotted from the results of experiments involving a tarred fiber 
driver and a type metal driven wheel show that when the slip is 2 per cent, the 
coefficient of friction developed under 150 pounds pressure per inch width is 
0.275 ; and under 400 pounds pressure per inch width, the coefficient of friction 
is 0.270. The maximum pressure at which tests of this combination of 
wheels were run was 400 pounds per inch width. 

LEATHER AND IRON 

Curves plotted from the results of experiments involving a leather driver 
and an iron driven wheel ^how that when the slip is 2 per cent, the coefficient 
of friction under a pressure of contact of 150 pounds per inch in width is 0.225 
and under a pressure of 400 pounds, 0.215. The maximum pressure at which 
tests of this combination of wheels were run was 400 pounds per inch width, 
although the leather driver was successfully operated up to a pressure of 750 
pounds per inch width. 

LEATHER AND ALUMINUM 

Curves plotted from the results of experiments involving a leather driver 
and an aluminum driven wheel show that when the pressure is 1 50 pounds per 
inch in width and the slip is 2 per cent, the coefficient of friction is 0.260; and 
when the pressure is 300 pounds per inch in width, the coefficient of friction is 
0.295. The maximum pressure at which tests of this combination of wheels 
were made was 350 pounds per inch width. 



276 AMERICAN MACHINIST GEAR BOOK 

LEATHER AND TYPE METAL 

Curves plotted from the results of the experiments involving a leather 
driver and a type metal driven wheel show that when the slip is 2 per cent, 
and the contact pressure 150 pounds per inch width, the coefficient of friction 
developed is 0.410. The greatest pressure at which tests of this combina- 
tion of wheels were run was 350 pounds per inch width. 

SULPHITE FIBER AND IRON 

Curves plotted from the results of the experiments involving a sulphite 
fiber driver and an iron driven wheel show that when the slip is 2 per cent, and 
the pressure 150 pounds per inch width, the coefficient of friction is 0.550. 
The maximum pressure at which tests of this combination of wheels were run 
was 350 pounds per inch width, although the sulphite fiber wheel 
was successfully operated up to a pressure of 700 pounds per inch width. 

SULPHITE FIBER AND ALUMINUM 

Curves plotted from the results of the experiments involving a sulphite 
fiber driver and an aluminum wheel show that when the slip is 2 per cent, and 
the pressure 150 pounds per inch width, the coefficient of friction developed 
is 0.410. The greatest pressure used in tests of this combination of wheels 
was 350 pounds per inch width. 

« SULPHITE FIBER AND TYPE METAL 

Curves plotted from the results of the experiments involving a sulphite 
fiber driver and a type metal driven wheel show that when the slip is 2 per 
cent, and the contact pressure 150 pounds per inch width, the coefficient of 
friction is 0.515. The maximum pressure used in tests of this combination of 
wheels was 350 pounds per inch width. 

• 

RESISTANCE TO CRUSHING 

Upon the completion of tests designed to disclose the frictional qualities of 
the several combinations, each fibrous wheel was subjected to test for the pur- 
pose of determining the maximum pressure per inch width of the face which 
could be sustained by it. This was accomplished by placing the wheel to be 
tested in the machine under a pressure of contact of 200 pounds per inch 
width. The load on the brake was then adjusted to give a 2 per cent, slip, 
and this brake load was maintained without change throughout the remainder 
of the tests. Thus adjusted, the machine was operated until the driver had 
completed 15,000 revolutions. This accomplished, and for the purpose of 
determining the reduction, if any, in the diameter of the fibrous wheel, the 
brake load was removed and the operation of the machine continued without 



FRICTION GEARS 277 

load for a period of 6,000 revolutions, the readings of the counters being taken 
at the beginning and at the end of the period. Under conditions of no load, 
the actual slip was assumed to be zero and the apparent slip observed was 
used for determining the reduction in diameter of the fibrous wheel which 
had been brought about by the previous running under pressure. This 
accomplished, the pressure of contact was increased, usually by 100-pound 
increments, and the whole operation repeated. This process was continued 
until failure of the fibrous wheel resulted. It will be seen that the ultimate 
resistance to crushing, as found by the process described, is that pressure 
which could not be endured during 15,000 revolutions. 
A summary of results is as follows: 

A CONCLUSION AS TO METAL WHEELS 

An examination of Table 25, which presents a comparison of values repre- 
senting the coefhcient of friction of the several combinations of wheels tested, 
reveals the fact that the relative value of the metal driven wheels is not the 
same when operated in combination with different fibrous driving wheels. It 
appears that those driving wheels which are the more dense work more effi- 
ciently with the iron follower than with either the aluminum or type metal 
followers; but in the case of the softer and less dense driving wheels, and 
especially in the case of those in which an oily substance is incorporated, 
driven wheels of aluminum and type metal are superior to those of iron. 
Finely powdered metal which is given off from the surface of the softer metal 
wheels seems to account for this effect, and the character ©f the driving wheels 
is perhaps the only factor necessary to determine whether its presence will be 
beneficial or detrimental. Finally, with reference to the use of soft metal 
driven wheels, it should be noted that no combination of such wheels with a 
fibrous driver appears to have given high frictional results. Except when 
used under very light pressures, the wear of the type metal was too rapid to 
make a wheel of its material serviceable in practice. 

CONCLUSIONS AS TO FIBROUS WHEELS 

The relative value of the different fibrous wheels when employed as drivers 
in a friction drive may be judged by comparing their frictional qualities as set 
forth in Table 25 and their strength as set forth in Table 26. The results 
show at once that the addition of belt dressing to the composition of a straw 
fiber wheel is fatal to its frictional qualities. The highest frictional qualities 
are possessed by the sulphite fiber wheel, which, on the other hand, is the 
weakest of all wheels tested. The leather fiber and tarred fiber are exception- 
ally strong; and the former possesses frictional qualities of a superior order. 
The plain straw fiber, which in a commercial sense is the most available of all 
materials dealt with, when worked upon an iron follower possesses frictional 



278 



AMERICAN MACHINIST GEAR BOOK 



qualities which are far superior to leather, and strength which is second only 
to the leather fiber and the tarred fiber. 



Sulphite Fiber 

Leather Fiber 

Straw Fiber 

Tarred Fiber 

Leather 

Straw Fiber with belt dressing. 



COEFFICIENT OF FRICTION WHEN CONTACT 
PRESSURE IS 150 POUNDS PER INCH 



IRON 


ALUMINUM 


TYPE METAL 


0-550 


0.530 


0.515 


0.515 


0.495 


0.350 


0.425 


0.455 


0.310 


0.250 


0.305 


0.275 


0.225 


0.360 


0.410 


0.120 







Table 25— Coefficient of Friction. 





load in 


decrease in 






POUNDS 


diameter 




Straw Fiber < 


200 
650 
750 


0.000 

0.053 
0.125 


I Wheel failed before running 15,000 revolutions 
under 750 pounds pressure. 


r 


200 


0.000 


■^ 




300 


0.005 






400 


0.013 






500 


0.021 




Leather Fiber - 


600 
700 
800 


0.027 
0.040 

0.051 


^ Wheel failed before running 15,000 revolutions 
under 1200 pounds pressure. 




900 


0.068 






1000 


0.099 






IIOO 


0.125 




>- 


1200 


0.200 


^ 


•• 


200 


0.000 


-| 




300 


0.026 






400 


0.038 






500 


0.052 




Tarred Fiber < 


600 
700 

800 


0.071 
0.098 

0.138 


^Wheel failed before running 15,000 revolutions 
under 1200 pounds pressure. 




900 


0.182 






1000 


0.250 






IIOO 


0.295 




^ 


1200 




^ 


' 


350 


0.047 


-< 


Leather 


450 
550 
650 


0.090 

0.015 

0.240 


^Wheel failed before running 15,000 revolutions 
under 750 pounds pressure. 


k. 


750 








200 


O.OIO 


■» 




300 


0.032 




Sulphite Fiber < 


400 

500 


0.056 
0.088 


^ Wheel failed before running 15,000 revolutions 
under 700 pounds pressure. 




600 


0.146 






700 


0.258 


-^ 



Table 26— Strength of Various Fiber Wheels. 



FRICTION GEARS 279 

THE POWER CAPACITY OF FRICTION GEARS 

A review of the data discloses the fact that several of the friction wheels 
tested developed a coefficient of friction which in some cases exceeded 0.5. 
That is, such wheels rolling in contact have transmitted from driver to driven 
wheels a tangential force equal to 50 per cent, of the force maintaining their 
contact. These wheels also were successfully worked under pressures of con- 
tact approaching 500 pounds per inch in width. Employing these facts as a 
basis from which to calculate power, it can readily be shown that a friction 
wheel a foot in diameter, if run at i ,000 revolutions per minute, can be made to 
deliver in excess of 25 horse-power for each inch in width. It is certainly true 
that any of the wheels tested may be employed to transmit for a limited time 
an amount of power which, when gauged by ordinary measures, seems to be 
enormously high; but obviously, performance under limiting conditions 
should not be made the basis from which to determine the commercial capac- 
ity of such devices. In view of this fact, it is important that there be drawn 
from the data such general conclusions with reference to pressures of contact 
and frictional qualities as will constitute a safe guide to practice. 

WORKING PRESSURE OF CONTACT 

The results of these experiments do not furnish an absolute measure of the 
most satisfactory pressure of contact for service conditions. Other things 
being equal, the power transmitted will be proportional to this pressure, and 
hence it is desirable that the value be made as high as practicable. On the 
other hand, it has been noted as one of the observations of the test that as 
higher pressures are used, there appears to be a gradual yielding of the struc- 
ture of the fibrous wheels; and it is reasonable to conclude that the life of a 
given wheel will in a large measure depend upon the pressure under which it 
is required to work. After a careful study of the facts involved, it has been 
determined to base an estimate of the power which may be transmitted upon 
a pressure of contact which is 20 per cent, of the ultimate resistance of the mate- 
rial as established by the crushing tests already described. This basis gives 
the following results: 

SAFE WORKING PRESSURES OF CONTACT 

PRESSURE 

Straw fiber 150 

Leather fiber 240 

Tarred fiber 240 

Sulphite fiber 140 

Leather 150 

COEFFICIENT OF FRICTION 

The coefficient of friction for all wheels tested approaches its maximum 
value when the slip between driver and driven wheel amounts to 2 per cent. 



28o AMERICAN MACHINIST GEAR BOOK 

and, within narrow limits, its value is practically independent of the pressure 
of contact. A summary of maximum results is shown by Table 30. In view 
of these facts, it is proposed to base a measure of the power which may be 
transmitted by such friction wheels as those tested upon the frictional quali- 
ties developed at a pressure of 150 pounds per inch of width, when operating 
under a load causing 2 per cent. slip. For safe operation, however, deductions 
must be made from the observed values. Thus, the results of the experi- 
ments disclose the power transmitted from wheel to wheel, while in the ordi- 
nary application of friction drives some power will be absorbed by the journals 
of the driven axle so that the amount of power which can be taken from the 
driven shaft will be somewhat less than that transmitted to the wheel on said 
shaft. Again, under the conditions of the laboratory, every precaution was 
taken to keep the surfaces in contact free of all foreign matter. It was, for 
example, observed that the accumulation of laboratory dust upon the surfaces 
of the wheels had a temporary effect upon the frictional qualities of the wheels, 
and friction wheels in service are not likely to be as carefully protected as 
were those in the laboratory. In view of these facts, it has been thought 
proper to use as the basis from which to determine the amount of power 
lirhich may be transmitted by such wheels as those tested, a coefficient of fric- 
tion which shall be 60 per cent, of that developed under the conditions of the 
laboratory. This basis gives the following results: 

COEFFICIENT OF FRICTION WORKING VALUES 

COEFFICIENT 
OF FRICTION 

Straw fiber and iron 0-255 

Straw fiber and aluminum 0-873 

Straw fiber and type metal o. 1 86 

Leather fiber and iron 0.309 

Leather fiber and aluminum 0.297 

Leather fiber and type metal 0.183 

Tarred fiber and iron o.i 50 

Tarred fiber and aluminum 0.183 

Tarred fiber and type metal 0.165 

Sulphite fiber and iron 0.330 

Sulphite fiber and aluminum 0.318 

Sulphite fiber and type metal 0.309 

Leather and iron 0.135 

Leather and aluminum 0.216 

Leather and type metal 0.246 

HORSE-POWER 

Having now determined a safe working pressure of contact and a repre- 
sentative value for the coefficient of friction, it is possible to formulate 



FRICTION GEARS 281 

equations expressing the horse-power which may be transmitted by each 
combination of wheels tested. Thus, calHng d the diameter of the friction 
wheel in inches, W the width of its face in inches, and N the number of 
revolutions per minute, the equations become, for combinations of, 

HORSE-POWER 

Straw fiber and iron 0.00030 dWN 

Straw fiber and aluminum 0.00033 dWN 

Straw fiber and type metal 0.00022 dWN 

Leather fiber and iron 0.00059 dWN 

Leather fiber and aluminum 0.00057 dWN 

Leather fiber and type metal 0.00035 dWN 

Tarred fiber and iron 0.00029 dWN 

Tarred fiber and aluminum 0.00035 dWN 

Tarred fiber and type metal 0.00031 dWN 

Sulphite fiber and iron •. . 0.00037 dWN 

Sulphite fiber and aluminum 0.00035 d^N 

Sulphite fiber and type metal 0.00034 dWN 

Leather and iron 0.00016 dWN 

Leather and aluminum 0.00026 dWN 

Leather and type metal 0.00029 dWN 

The accompanying chart gives a convenient means of determining the 
value of any one of the variable factors in the formula horse-power = 0.0003 
dWN for the straw fiber friction wheel working in combination with an iron 
follower, the remaining factors being known or assumed. To transform 
values thus found to corresponding ones for the other possible combinations 
of wheels, it is necessary only to multiply by the proper factor chosen from 
the table of multipliers given with the chart. 

APPLICATION or RESULTS TO FORM OTHER THAN THOSE EXPERIMENTED 

UPON FACE FRICTION GEARING 

A fibrous driving wheel, acting upon the face of a metal disk, constitutes a 
form of friction gear which is serviceable for a variety of purposes. If the 
driver is so mounted that it may be moved across the face of the disk, the 
velocity ratio may be varied and the direction of the disk's motion may be 
reversed. The contact is not one of pure rolling. If the driver is cylindrical 
in form, the action along its line of contact with the disk is attended by slip, 
amount of which changes for every different point along the line. The recog- 
nition of this fact is essential to a discussion of the power-transmitting 
capacity of the device. 

Experiments involving the spur form of friction wheels already described 
have shown that slip greatly affects the coefficient of friction; that the co- 
efficient approaches its maximum value when the slip reaches 2 per cent., and 
that when the slip exceeds 3 per cent., the coefficient diminishes. It is 



282 AMERICAN MACHINIST GEAR BOOK 

known that reductions in the value of the coefficient with increments of sUp 
beyond 3 per cent, are at first gradual, although the characteristics of the test- 
ing machine have not permitted a definition of this relation for slip greater 
than 4 per cent. The experiments, however, fully justify the statement that 
for maximum results the slippage should not be less than 2 per cent, nor more 
than 4 per cent. It is the maximum limit with which we are concerned in 
considering the amount of power which may be transmitted by face friction 
gearing. 

From the discussion of the previous paragraph, it should be evident that, 
for best results, the width of face of the friction driver and the distance 
between the driver and center of disk should always be such that the variations 
in the velocity of the particles of the disk having contact with the driver will 
not exceed 4 per cent. A convenient rule which, if followed, will secure this 
condition is to make the minimum distance between the driver and the center 
of the driven disk twelve times the width of the face of the driver. For 
example, a driver having a 3^^-inch width of face should be run at a distance 
of 3 inches or more from the center of the disk. Similarly, drivers having 
faces 3^^, I, or 2 inches in width should be run at a distance from the center of 
the disk of not less than 6, 12, or 24 inches, respectively. When these condi- 
tions are met, all formulas for calculating the power which may be transmitted 
apply directly to the conditions of face driving. 

It may not infrequently happen that friction wheels must be run nearer 
the center of the disk than the distance specified; there is, of course, no objec- 
tion to such practice, but it should not be forgotten that as the center of the 
disk is approached, the coefficient of friction, and consequently the capacity 
to transmit power, diminishes. 

CONDITIONS TO BE OBSERVED IN THE INSTALLATION OF 
FRICTION DRIVES 

Whatever may be the form of the transmission, the fibrous wheel must 
always be the driver. Neglect of this rule is likely to result in failure which 
will appear in the unequal wear of the softer wheel, occasioned by slippage. 

The rolling surfaces of the wheel should be kept clean. Ordinarily they 
should not be permitted to collect grease or oil, nor be exposed to excessive 
moisture. Where this cannot be prevented, a factor of safety should be pro- 
vided by making the wheels larger than normal for the power to be trans- 
mitted. 

Since the power transmitted is directly proportional to the pressure of con- 
tact, it is a matter of prime importance that the mechanical means employed 
in maintaining the contact be as nearly as possible inflexible. For example, 
arrangements of friction wheels which involve the maintenance of contact 
through the direct action of a spring have been found unsatisfactory, since 
any defect in the form of either wheel introduces vibrations which tend to 



FRICTION GEARS 283 

impair the value of the arrangement. It is recommended that springs be 
avoided and that contact be secured through mechanism which is rigid and 
which when once adjusted shall be incapable of bringing about any release 
of the pressure to which it is set. 

EXPLANATION OF CHART 

Chart 15 is plotted for the most common materials used for friction 
gearing, straw fiber and cast iron, and gives means of determining the variable 
factors for the fiber wheel in the formula horse-power = 0.0003 dWN , in 
which d is the diameter of the wheel in inches, W its width of face in inches, 
and N the number of revolutions per minute. 

To use the chart for other friction materials multiply the values obtained 
from the chart by the proper factor selected from the table below: 

Straw fiber and aluminum i.io 

Straw fiber and type metal o • 73 

Leather fiber and cast iron i . 97 

Leather fiber and aluminum i . 90 

Leather fiber and type metal 1.17 

Tarred fiber and cast iron o • 97 

Tarred fiber and aluminum 1.17 

Tarred fiber and type metal i . 03 

Sulphite fiber and cast iron i . 23 

Sulphite fiber and aluminum 1.17 

Sulphite fiber and type metal i . 13 

Leather and cast iron 053 

Leather and aluminum 0.87 

Leather and type metal o • 97 

(a) To find the total horse-power which can be transmitted by a wheel, 
having given the diameter of the wheels in inches, the width of its face in 
inches and the revolutions per minute, locate the intersection of the vertical 
line representing the given speed with the diagonal line representing the given 
diameter. Follow the horizontal line passing through this point, to the right 
or left as the case may be, until it intersects the vertical line representing 
the given width of the face. The diagonal line through this point will give 
total horse-power required from the scale so marked. 

{h) To find the speed in revolutio?is per minute for a wheel, having given its 
diameter in inches, its width of face in inches, and the total horse-power to be 
transmitted, locate the intersection of the vertical line representing the width 
of face with the diagonal line representing the total horse-power to be trans- 
mitted. Follow the horizontal line passing through this point, to the right or 
left as the case may be, until it intersects the diagonal line representing the 



284 



AMERICAN MACHINIST GEAR BOOK 



diameter in inches. The vertical hne passing through this point indicates on 
the scale at the bottom of the chart the speed required. 

(c) To find the width of face in inches for a wheel, having given the total 
horse-power to be transmitted, its diameter in inches and its speed in revolu- 
tions per minute, locate the intersection of the vertical line representing the 
given speed with the diagonal line representing the given diameter. Follow 



Face, "Width in Inches 




Speed, Revolutions per Minute 



la w t> go o> 



CHART 15. PROPORTIONS OF FIBROUS FRICTION GEARING. 

the horizontal line passing through this point, to the right or left as the case 
may be, until it intersects the diagonal line representing the given total horse- 
power. The vertical line passing through this point will indicate the width 
of face required on the scale at the top of the chart. 

{d) To find the diameter in inches for a wheel, having given the horse-power 
to be transmitted, its width of face in inches, and its speed in revolutions per 
minute, locate the intersection of the vertical line representing the width of 
face with the diagonal line indicating the total horse-power. Follow the hori- 
zontal line passing through this point, to the right or left as the case may be, 



FRICTION GEARS 285 

until it intersects the vertical line representing the speed. The diagonal line 
passing through this point represents the diameter which is required. 

(e) To find the surface speed of a wheel, having given its diameter in 
inches and its speed in revolutions per minute, locate the intersection of the 
vertical line representing the speed in revolutions per minute with the diagonal 
line representing the given diameter. The horizontal line passing through 
this point represents the surface speed in feet per minute which is required, 
and which is read on the vertical scale at the right of the chart. 

(/) To find the horse-power per inch of face for a wheel, having given the 
total horse-power transmitted and the width of the face in inches, locate 
the intersection of the vertical line representing the width of face with the 
diagonal line representing the total horse-power. The horizontal line passing 
through this point represents the horse-power per inch of face required and 
may be read on the vertical scale at the left of the chart. 

FRICTION DRIVE ON A FORTY-FOUR FOOT PIT LATHE* 

The machine here described was designed to meet the demands of an 
establishment manufacturing the heaviest type of electrical machinery. The 
ever-increasing dimensions of this class of machinery make it particularly 
desirable that the existing heavy machine tools should be capable of exten- 
sion of capacity with a view to probable future requirements, and that a pit 
lathe is peculiarly adapted to such extension will, doubtless, be readily 
admitted. 

The face-plate of this machine measures 30 feet in diameter, and the pres- 
ent dimensions of the pit will admit of swinging 44 feet on* centers, with a 
maximum width of 12 feet. The large face-plate is built up of twelve seg- 
ments. The rim is of box sertion, the ends of the rim in each section being 
finished to make the joint, and the segments being held together at the rim 
by body-bound bolts. The arms are slotted for bolts, and the space between 
segments is also shaped to receive the usual square-headed bolts, as the inner 
end of each segment is fastened to the smaller face-plate by several body- 
bound bolts. 

A feature of interest in connection with this machine is the method of 
drive adopted, which is a friction roller, 18 inches diameter, made of com- 
pressed paper, while the rim of the large face-plate, 15 inches wide, affords 
the necessary contact surface for driving. 

Power is supplied by a 75 horse-power motor, quadruple-geared, the use 
of the multiple voltage system giving the machine a range covering all diam- 
eters from 6 feet to the present capacity, though the gear train is designed 
to admit of two changes of back gear in addition. 

* Extract from a paper presented at the New York meeting of the American Society of 
Mechanical Engineers by John M. Barney. 



286 



AMERICAN MACHINIST GEAR BOOK 



Fig. 246 shows the assembled pit lathe driven by the friction roller while 
taking a heavy facing cut, on which occasion four tools were employed. The 
picture also shows the driving motor with its train of gears and the mechanism 
employed for adjusting the pressure on the friction roller. 




FIG. 246. rRICTION-DRrVEN LATHE. 



AN INVESTIGATION OF FIBER FRICTIONS 

The Kelsey Motor Company, during 1921, conducted an exhaustive 
investigation of the properties and behavior of friction drives, the results of 
which were presented in a paper read before a meeting of the Friction Drive 
Engineering Society by Philip Kriegel.* Abstracts from this address are: 

''In order that a comprehensive idea might be secured of the functioning 
of the friction fibers under different conditions of speed, slip and twist, the 
friction wheels were brought in contact with the disk at four different posi- 
tions with regard to the disk. These were so placed that the middle of the 
working face of each fiber was at approximately 7^ inches, 6 inches, 4 inches, 
and 2 inches from the center of the dish. With the motor running at about 
1,625 revolutions per minute, as was the average throughout the test, the 
speeds in the four positions would be, at an assumed zero slip, about 

1480 r. p. m. or 6,600 feet per minute 
1 140 r. p. m. or 5,100 feet per minute 

760 r. p. m. or 3,400 feet per minute 

380 r. p. m. or 1,700 feet per minute 

Beginning with a contact pressure of 250 feet a load was sent through 
the transmission, the most that could be absorbed without excessive slipping. 

*Experimental Engineer, Kelsey Motor Co. 



FRICTION GEARS 



287 



A complete record was taken of the horse-power input, the speed of both 
disk and friction wheel and the torque on the dynamometer scale. Pressures 
increasing in increments of 25 pounds from 250 to 500 pounds, and 
increments of 50 pounds up to 1,050 pounds were applied. The runs in each 
position were repeated. 

A general idea of the type of fibers investigated, their composition, 
characteristics and dimensions can be gained from Table 27, The disk, 17 
inches in diameter, used for Fibers No. i to No. 6, was of "amelite" (an 
aluminum-zinc-copper alloy). For Fibers No. 7 to No. 10 it was found 
necessary to face the disk from the periphery inward 4 inches with boiler plate 
steel. 

Table 27 





DIMENSIONS, 




INCHES 


FIBER 
NO. 


WIDTH 
OF 


OUTER 




WORK- 
ING 
FACE 


DIAM. 

OF 
FILLER 



TYPE OR NAME 



COMPOSITION AND REMARKS 



2 
3 

4 
5 
6 

7 
8 

9 

10 



1-55 


16.95 


1-55 


16.80 


1-55 


16.80 


1-35 


16.95 


1.50 


16.95 


1 .60 


17.00 


1.30 


16.95 


1.30 


17.00 


1-45 


16.95 


1.50 


16.95 



"Tarred" fiber 

Cotton duck fiber 

Metal and brake 
lining fiber 
"Micarta" fiber 

Black split fiber 

Paper fiber 

Molded fiber 

"Egyptian" fiber 

"Oil celoron" fiber 

"Chrome friction" 
fiber 



Paper — Kraft — 100% — tarred, compressed 

in rings 
Radial pieces of cotton duck shaped into a 

ring under high compression 
Strips of brake lining (containing asbestos) 

in alternate rings with copper mesh 
A chemical — (bakelite?) laid in rings be- 
tween layers of cotton gauze 
Paper— 100% rag^ — chemically treated; 

made in two split rings 
Paper — rag — 35%; sulphite 25%, rope 40%. 

Very soft. Improperly finished 
Believed to be molded asbestos which had 

probably been treated with cement 
A rag fiber, chemically treated, vulcanized, 

and hardened under pressure 
Paper containing rag — 5%; sulphate — 95%. 

Treated with oil and compressed 
Rawhide, presumably treated with chrome 

solution 



In Table 28 are shown the average horse-power transmitted by the fibers 
in each of the four positions indicated, the speeds of the friction wheel, and 
the corresponding road mileage at a gear reduction of 4.5 to i (now used in the 
Kelsey automobile). Every effort was made to transmit the same or as much 
horse-power as possible at the minimum contact pressure apphed and to keep 
the horse-power input fairly constant with increasing pressures. The varia- 
tions in the amount of horsepower transmitted must give rise to certain- 
indications which will be discussed later. 



288 



AMERICAN MACHINIST GEAR BOOK 











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ON 






















M 












M 00 


lO VO 


00 M 


Ov Ov 


t^ o^ 


vo r^ 


VO t^ 


CO r^ 


^ 


Ol 








M 




NO fC 


vO W 


VC C^! 


VO H. 


VO M 


VO M 


NO M 


NO M 


VO M 


M 


t^ 




t^ o 


VO C^ 


00 o 


O t^ 


VO -^ 


O LO 


O <N 





'd- O 


^S 








O^ 






















C/3 




O 








r^ 't 


O ^ 


O M 


d Ov 


00 On 


00 Ov 


00 ON 


r^ On 


VO Ov 


?^ 






PO 




00 M 


Ov M 


Ov M 


Ov 


00 


00 


00 


00 


00 


<ll 


t^ 


<N 1/ 


■5 O lO 


O t^ 


VO t^ 


rrvO 


CO r^ 


O VO 


Tt VO 


M VO 


CM VO 


P^ 






Ov 






















H 










OnvO 


O) lO 


(N rf 


O '^ 


O 't 


O Tt- 


O '^ 


On '^ 


Ov rt 


r-. Tf 












00 


ON 


ON 


Ov 


ON 


ON 


ON 


00 


00 


00 


H 


rOOC 


t^ UO 


O <N 


O VO 


Tl- O) 


t^ 


O <v) 


O) O 


VO <N 


O l-H 


IZ 








t^ 






















M 










6 0* 


:> fo fo 


ro PO 


VO CO 


VO CO 


oi CO 


O) CO 


O CO 


Ov CO 


Ov CO 


O 








r^ 


On 


0^ 


Ov 


Ov 


ON 


Ov 


Ov 


Ov 


00 


00 


O 

n 






g 




























































w 








































































w 
ft 


>. 


X 




>. 




>. 




>. 




>. 




>. 




>> 




>. 




>. 




rCt 






O 


u 




U 




o 




u 




u 




u 




u 




o 




u 




(S 






C 


c 




C 




c 




c 




a 




C 




c 




C 




c 












m 


<u 




O) 




<v 




<u 




<v 




<v 




<u 




<u 




<u 




P3 




















































u _ 


u 


(J -: 


^ _■ 


<-> ^ 


u -." 


^ _: 


w _■ 


u «■ 


<-> ^ 








^•i 


^ bg.S- 


^.s- 


hE.S- 


bB-B- 


Jfi.S- 


e.^ 


^.^ 


S5.B- 


bB-S- 


< 








WcT: 


W c75 


Hc73 


Wc75 


Wc75 


Wc^ 


Wc75 


Wc/D 


Wc/i 


W GO 




W 
(A 


H" SS 
























CO f^ 


u p 


o 


o 


o 


VO 


o 


VO 


o 


VO 


o 


O 




< s 


lO 


o 


VO 


t^ 


o 


M 


VO 


t^ 


o 


VO 




oi O 


fe! t^ 


«N 


PO 


fO 


CO 


"* 


''t 


■^ 


■^ 


VO 


VO 




M 


S o 
























Pi 


,9 ^ 


























Pi 


u 


1 










1 






1 






1 






1 






1 






1 






1 






1 






1 



FRICTION GEARS 



289 



OC 


3 OC 


3 r^ 10 00 H 


1 vc 


H 


t^ 1 to r^ 1 ■* 


CO t/ 


3 t^ !>. 


■^ ^^ 


H M C 


) 00 M VO M 10 <N 


'^ CV 


CN) CJ 


Ov r 


1- 00 c 


:i t^ CO 


t^ 1- 


< t^ (- 


t vOm vOm vOi-i|vOh 


VO H- 


10 H 


to 1- 


to H 


c 


) ONOC 


) 00 u 


Ovto OvO vovo Oto Ot' 


•3 t^ 


■^ to 


00 ^ 


t--M: 


) VO vc 


> lioto tovo toTh toTf to^ 


h M -!! 


h On CO 


00 >- 


< 00 1- 


< 00 H 


< OOM OOM COM OOi- 


1 00 H 


00 H 


t^ M 


10 c 


c 


M u 


-) VO c 


Ov >- 


00 1- 


00 1- 


to ■<; 


h to tr 


■3 Tj- 


H 1- 


< ro t- 


M C 


Cv c 


t^ c 


vO c 


vo C 


to C 


to 


rh 


H 


0> H 


c> >- 


00 H 


00 H 


00 H 


00 1- 


00 H 


00 H- 


00 H-l 


10 c 


> c 


) <N C 


) 00 vc 


> loO MO too >-> <^ 


3 t/ 


■3 VO CO 


rO C 


> rO C 


^ ro c 


^ <N OC 


3 M 00 pooo <N 00 <N ^ 


t- 


t^ 





0- 


On 





On 


1 


1 On 


1 


On 





r<o ^ 


"^^ 


) vC 


3 <M OC 


3 00 1^ On C> tovc 


3 CJ OC 


M to 


00 


\ vo ^ 


» VO ^ 


< to ^ 


) rj-CNj COCJ CO^J cocv 


CO H 


) H 


vo >- 


VO H 


J VO H 


■1 VO H 


■1 vOm vOm vO»h vOh 


VO H 


VO M 


«N 1/ 


■5 On i^ 


■) CS 1/ 


■> VO f 


-> vo 1/ 


-) ^1- c 


t-- c 


c 


00 C 


to 


On^C 


I^VC 


t-^vC 


'^vc 


cove 


cove 


W vc 


CJ vc 


vc 


00 to 


00 


00 


00 


00 


00 


00 


00 


00 


00 


r^ 


'=; 


1- <N OC 


^ 


t- CO c 


vO c 


■3 to "^ 


(- CO to VO tr 


■3 CO C 


CO C) 


H 1/ 


■> VO ■<; 


h 10 t; 


h '* -^ 


h CO '^ 


l- w T 


1- C4 "^t '^ 


1- -s 


1- Tj- 





00 


00 


00 


00 


00 


00 


1 00 


00 


00 


IT) U 


•5 H C 


M C 


CJ C 


Cs 


'^ c 


CO C 


1 00 vc 


00 ti 


■3 rO 


M U 


■) Ml/ 


■5 ^ 


-) 00 t/ 


■5 00 t^ 


■5 t^ u 


■3 t^ to 1 VO T 


1- VO T 


1- r^ '^ 


On 


On 


Ov 


00 


00 


00 


00 


1 00 


00 


CO 


'^ J> 


t- 


J> 


^■ 


- 1 t^ t- 


~ CJ c 


■3 1^ 1 CO c 


■3 CN) t~ 


CO 


<N rj 


h <N -^ 


h ^ 


1- '^ 


1- C> Tl 


1- ■>; 


h 00 ^ 


1- 00 'J 


1- 00 ^ 


1- t^ '^ 


r^ H 


t^ h- 


t~~ (- 


t^ h- 


VO t- 


VO t- 


VO 1- 


vO ^- 


VO H 


VO M 


r^ ly 


:> CN OC 


^vc 


to C 


K to 


00 


M "^ 


h 01/ 


■3 CO 1> 


to to 


rt <N 


10 OC 


vO t" 


c^ cs 


-* 


tooc 


r^ t: 


1- VO c' 


"3 to CJ 


CO C) 


r^ r/ 


■5 r^ (N 


t^ OJ 


00 <N 


00 01 


CO H- 


00 >- 


CO 1- 


00 H- 


00 M 


"^ 


h 10 H 


CO H 


M C 


:> 00 -^ 


f- 0\ c 


■3 rj- 


'^ c 


00 


ON 


00 Tj 


1- 00 rj 


h t^ -^ 


h t^ ^ 


h VO -"^ 


1- to r; 


h ^ 'Si 


t- ^ T 


1- CO Ti 


1- C^ Tf 


00 


00 


00 


00 


00 


00 


00 


00 


00 


00 





^ 


to 


tooc 


to 


M \J 


■3 00 VC 


t> 


■3 VO 1/ 


■3 T^ -* 


00 ^ 


•) 00 •J' 


:) r^ 1/ 


■> t^ -si 


h r^ u 


■3 l^ T 


t- VO t: 


1- VO ^ 


h to t; 


h to Tt 


00 


00 


00 


00 


00 


00 


00 


00 


00 


00 


-^ 


h ^ 


h to 


CM 


n 


(■ to '^ 


f t^ H- 


r^ 1- 


CO 1- 


CO M 


CN 


10 cs 


CO p' 


■5 c?v c 


3 00 <N 


t^ Cv 


'^ •- 


Tt- 1- 


Tt 1- 


CJ H 


H 


IT) H 


to H 


"* M 


Tl- H 


-'d- t- 


Th H 


rj- h- 


n- 1- 


T^ M 


10 1/ 


■> 00 


CO 


1 C^ t- 


M 1/ 


■3 <N C 


^ 


M C 


c 


On Tt 


f 


•> 00 P' 


-> 00 c 


■3 1 t^ 1- 


f~~ 1- 


NO 1- 


to >- 


'^ 


CJ 


M Ov 


00 H 


t^ h- 


r^ 1- 


1 ^^ ^ 


f^ 1- 


t-^ 1- 


S>~ t- 


!>. H 


r^ 1- 


t-^ 


H f 


:i vO r* 


■3 to H- 


10 Ti 


1- 


CO c 


10 OC 


•- 


tO h- 





t> 


t^ t- 


t^vc 


VO vc 


VO t~ 


10 r- 


Tl- t/ 


■3 Tt-vC 


covC 


CO to 


00 


t-^ 


r>. 


t^ 


f^ 


1^ 


t^ 


t^ 


r^ 


t^ 


CO 'd 


|- (N ly 


■) 00 <N 


CJ 


Tf w 


OV H 


On CJ 


H C 


•CO tc 


■3 to 


vO iJ" 


■> vO 1/ 


•) CO t/ 


■) CO !/■ 


1 1-1 1/ 


3 Ch t/ 


■) t^ 1/ 


3 t^ 1/ 


■3 to ^ 


h ^ ^ 


00 


00 


00 


00 


00 


f- 


r^ 


t^ 


r^ 


r^ 


COOC 


rOOC 


t-> c 


^ C 


CNi 


to C 


■3 M t; 


t- ^ 


■3 CO t- 


VO VO 


O^ <N 


00 cs 


t^ CN 


VO c 


■) to c< 


■3 CO c< 


■3 ^ 


h M t; 


(- 00 c 


■3 J>. CO 


u-> h- 


VO H- 


to H 


to H- 


to H 


to H 


to l- 


to 1- 


•>* h- 


Tj- M 


J^ 


H 


OnOC 


cs u 


■5 t^ C 


OC 


't 


't C 


CJ OC 


00 


10 C 


^ rO C 


K M OC 


<N OC 


M OC 


M ^ 


OC 


00 OC 


f^ t- 


NO 00 


00 


00 


00 


00 


00 


00 


00 


f^ 


t^ 


1^ 


00 ^ 


■> vO I' 


■) r>. 1/ 


■3 -^ t/ 


1 VO t» 


■3 VO r- 


t- 


CI C 


00 c 


CO CO 


to Tj 


1- 10 Ti 


1- -* '^ 


h CO t; 


h cs t; 


1- M t; 


t- M t; 


h Of 


■3 00 1/ 


■3 00 to 


OC 


00 


00 


00 


00 


00 


00 


00 


t^ 


r^ 


00 '^ 


1- 10 Tj 


1- Ov c 


■> 00 <N 


1 vc 


vC 


f- 


vc 


00 vc 


00 VO 


r>. c* 


■) r^ cv 


■) U-) c< 


~> to c 


■3 't c 


■3 CO c< 


■3 CO c< 


•3 CO c< 


■3 M C 


•3 M CO 


00 


00 


00 


00 


00 


00 


00 


CO 


00 


00 


> 


% 


> 


-> 


> 


% 


> 


■> 


> 


% 


> 


■^ 


> 


■» 


> 


1 


> 




> 






u 


u 


u 


(J 


CJ 


CJ 


CJ 


CJ 





CJ 




C 


a 


C 


d 


a 


C 


G 


C 


c 


c 




<u 


(U 


dj 


D 


<u 


<u 


<v 


(U 


<u 


03 


























(J ^ 


CJ ^ 


u ^ 


CJ ^ 


^ 


u ^ 


CJ _ 


^ 


CJ 


CJ 




ifi.t: 


^ ^.£ 


^ h6.£ 


^ hg.S 


^ sfi.t 


^ ^.b 


"^ b6.£ 


'^ ^s 


^ ^.fc 


^ sfi.S- 


He/: 


H cc 


WcT: 


Hcc 


WcT: 


WEc 


WcT: 


WcT: 


Hoc 


H CO 



































LO 





to 





to 





to 





to 


VO 


VO 


t^ 


t^ 


00 


00 


On 


On 










































M 




H 





290 



AMERICAN MACHINIST GEAR BOOK 


















t~^ t^ 


ON 00 


CO <N 


CO On 


On On 


On m on On On 


ON On 










t^ 


^ 


Tl- 


rO M 


CO H 


■^ M CNI tH M HH 


M M 








On 




t^ (N 


t^ (N 


t^ <N 


t^ (N 


l^ C^ 


r^ <N r^ <N t^ M 


t^ CN 




VO M 


>o 


'si- 











^ to 00 to 


00 




w 

H 




10 t^ 


t-- 


CO On 


Tf On 


Tt On 


rj- ON 


rj-oo CN 00 cq t^ 


CN t^ 






00 

ON 


t^ W 


J>. M 


00 


00 


00 


00 


00 


1 00 


1 00 


00 




VO 


^ 


'^ rf 


'!^ 


r-* M 


t^ H 


vO H On M On M 


On 00 








vovo 


t^NO 


VO NO 


■^NO 


CN) 


(N VO 


CN) NO M VO M NO 


to 








10 


00 


00 


00 


00 


00 


00 


00 


1 00 


1 o< 


3 
Tn 


« 
C 


3 







^ 


t>. CO 


00 


00 


00 


NO to C 


) 










NO t^ 


t--NO 


f-^vO 


t^vo 


t>.vO 


i^NO 


t^vo vO vO tovO 


'^NO 











00 


00 


00 


00 


00 


00 


00 


1 00 


1 oc 


00 













LO LO 


CN) XT) 


rt- rj- 


■^ to 


CO CM 


. . 1 . . 1 . . 

10 00 00 


6^^ 




W 






<N IN 


<S CNJ 


10 CO 


CO H 


<N M 


M H MM M 


M 




1— 1 




4 




vO CO 


00 <N 


00 M 


00 M 


00 M 


00 M 00 M 00 M 


CO M 







On 


00 


CN 00 


H C50 


NO to 





NO 00 00 C3n 


00 00 






W 




10 


<N 10 


CO 10 


c>^ 10 


CN to 


CO to 


CO to CO ■* CO CO 


CO CO 






12; 


00 


On 


ON 


ON 


On 


On 


ON 


ON 


1 On 


1 ON 


On 




<N 


10 m 


VO t^ 


rh CO 


NO 


r^ 


NO VO NO to •^ to 


to 










NO On 


f- 


VO 


NO VO 


to NO 


tOVO 


to to to to to to 


to to 






t^ 


00 


ON 


On 


00 


00 


00 


00 


1 00 


1 00 


00 






ON 






CN On 


'^ 


11 


NO 


VO 1 lovo 1 t^ M 


NO 






Q 



















• 


■ • • 








fe 










CN 


to M 


On CO 





d 1 00 1 VO 


to f^ 






<N 






t^ CS 


t^ <N 


t^ M 


00 H 


00 M j r^ M 1 f-~ M 


t^ 




On 




10 <N 


to CN| 


to CNI 


CO 


00 


00 1 f^ 1 to 


':!- 






»; 



















• 


. . . 






H 


w 






CO CO 


to 00 


r^ M 


d 00 


M t^ 


t^ 1 t^ 1 t^ 


ON Tj- 






M 

w 


H 

W 

u 


4 




VO 't 


t^ (N 


00 CN 


On m 


On M 


On M 1 On M 1 On m 


00 M 




ON 





<N Tt 


rt CO 

d d 


'^ 

d d 


M 
(N 


"st- 

cs 


J>- VO 00 CO 
00 CX3 On CX) On 


»o 

t^OO 









vo 




00 H-i 


On m 


On w 


On m 


On m 


On M 1 00 


1 00 


00 


« 


>0 


10 -^ 


10 10 


to <N 


to 


00 


f^ -^ 1 to 1 to 


CO 


w 




pi 


00 
















• 


• • • 




CO 






rj- ON 


10 <o 


CO to 


<N 


CO 


to On 


Th On 1 '^ On 1 CO On 


CO 00 


§ 




r-« 


00 <N 


00 <N 


00 M 


On m 


On m 


On 


On 


1 


1 





t^ 




On 




Tj- CN 


VO 


to CO 


VO 00 


M CO 


00 C 


> NO 1 VO 


^ M 


y, 




W 





















• • 















t^ 


CN| NO 


to -"^ 


On m 


M 


M C 


(N On 1 <N On 


00 Ov 


« 


^ 


(N) 

o\ 





J^ M 


t^ M 


t^ M 


1^ M 


00 M 


00 H 


^ 00 


1 00 


r-^ 


w 






10 H 


<N) 

4 On 


to 

Tj-VO 


00 to 

<o "* 


1>» CO 

CO "^ 


NO C 

CO -^ 


) to C 

1" CO T 


) NO 

j- CO 4 


CN CO 


C/2 


P=^ 

Q 


On 





On m 


ON 


On 


ON 


On 


On 


On 


On 









00 <N 

'^NO 


CO <N 
to to 


^ 
to to 


M M 
Tj- CO 


00 00 
CO CN 


COOC 

CN] CN 


M CN 


CN 


^ 
t^ CO 






hi 


VO 




_C 
u 


> 


ON 


On 


On 


On 


On 


On 


00 


00 




00 10 


ovo 


CO 


to 


CO CM 





C 


) NO i- 


•"^ to 


On 






Ph 


W 


00 


rr) ir> 


t^ CO 


VO CM 


to CN 


to CS 


to <N 


to ^ 


Tl- H 


Tj- M 


CO CN 






I^ 


On 


On 


On 


ON 


On 


On 


ON 


On 


On 


ON 






ON 








CO 
4 




VO 
CO 




<N 
CO 




CO 
CO 




to 

CO 




NO 
CN 


00 
M 


d 





■ d 








< 
H 


H 








r-» 




I^ 




^>. 




r^ 




r-« 




t-- 


t^ 


t-* 


— 


t^ 






tT) 























<N 




(M 




to 


CN 


to 


00 










On 












































Q 










(s 




to 




Tt- 




CO 




(N 




M 


M 







On 










rn 








00 




00 




00 




00 




00 




00 


00 


00 




t^ 






ro 








t^ 




00 




M 




On 




'-' 




ON 


to 


Tl- 




(N 












M 




<N 




<N 




CN) 




00 




00 




to 


Tj- 


CO 




M 










10 


On 




On 




ON ■ 


On • 


00 • 


00 • 


00 


00 


00 




00 






00 


1>.00 


00 


<N 


«>■ 


•^ 


M 


c 


t^ c 


t~«-vO 


NO to 










u-)00 


NO 00 


00 t^ 


NO l^ 


to t^ 


to r^ 


rj t> 


CN r~ 


(N to 


M to 








r^ 


00 


00 


00 


00 


00 


00 


00 


00 


00 


00 






u 


> 






> 






> 






> 


■^ 




> 


% 




> 


■> 




> 


% 


> 


"» 


> 


~> 




> 


■> 








u 




u 




(J 




CJ 




a 




CJ 




CJ 


<j 


CJ 




CJ 








C 




c 




C 




C 




C 




C 




C 


C 


c 




c 










<u 




<u 




(U 




(U 




<u 




flj 




<v 


OJ 


OJ 




<u 




















































u 









CJ 




■ 


(J ^ 


CJ 


(J ^ 


CJ ^ 


CJ 


CJ ^ 








Effi 
Slip 




Wc75 


Wc75 


Effi 
Slip 


Effi 
Slip 






Effi 
Slip 






M 


^ 
























(A 


H tn 
























p 


^ 











to 





to 





to 










tfl Pm 


^ ^ 


<o 





to 


t^ 





<N 


to 


i^ 





to 




en 


S t) 


Cl 


CO 


(O 


CO 


Tt- 


'^ 


-i- 


^ 


<o 


to 




■4 


5 
























« 


,9 PM 
























^ 





1 




































1 














1 






1 



FRICTION GEARS 



291 



ON 


On 


00 
00 


00 

00 

t^ t-c 


t^NO 

00 00 
\0 *-* 


NO On 

NO NO 

vO i-i 


00 On 

to NO 

\0 M 


t-^ On 

TtNO 
NO M 


VO On 

M NO 
VO M 


CO 00 

NO 

w 


»o 

00 




d !>. 

. 00 


10 

NO 

00 


CO 10 

00 NO 


CO 

t^NO 


00 
to NO 


<N to 

rf to 


M 
^ to 


c 

c 
r 


) 
to 

-4 00 

t to 


c 

c 
f 

~NC 

c 

r 


^ 

^ 10 


t^ 

10 

00 


t^ 

10 

00 


M 10 

On 10 


r^ 10 

00 10 


00 NO 


CO 
t^ 10 


NO to 


NO to 
rt- to 


c 


3 04 

to 


00 

00 


10 

fONO 

00 


M 

PO 10 

00 


1 

rr> to 

1 00 


00 

t-l LO 

00 


to 

M 10 

00 


'^ 

00 


CO 

M Tt 

00 


NO 04 

'^ 

00 


to 

On CO 






















t^ 

ON M 


10 NO 

On 

t^ M 


00 
tc. 


10 t^ 

4 d 


00 
CO 


to 

d M 


04 to 

d d 


^ 
d d 




d On 


t^ 

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NO 


o\ 


On 

00 -^ 
00 


On NO 

00 Tj- 
00 


CM 

00 "^ 
00 


M CO 

00 


^ 

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00 


CO NO 
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00 


NO 

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00 


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CN) 10 

00 


00 

M 10 

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M ^ 

00 


CM NO 

M Tl- 

00 


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00 


6 t^ 


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NO 


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l^ to 

NO 


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NO 


CO 

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10 


1^ 

to 


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01 10 

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Lo 

00 M 




t^ CO 

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NO 

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00 to 
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CO to 
to M 

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00 M 


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1000 

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00 


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1 



292 



AMERICAN MACHINIST GEAR BOOK 

TABLE 29 



FIBER 
NO. 


MIDDLE 

OF 

FIBER 

IN 
CON- 
TACT 
WITH 
DISK 
AT INCH 


AVER- 
AGE 
HORSE 
POWER 
TRANS- 
MITTED 


AVER- 
AGE 
SPEED 

OF 
FRIC- 
TION 
WHEEL- 
R.P.M. 


AVER- 
AGE 

MILES 
PER 

HOUR 

(gear 

REDUC- 
TION 
4.5 TO 
I) 


FIBER 
NO. 


MIDDLE 

OF 

FIBER 

IN 
CON- 
TACT 
WITH 
DISK 
AT INCH 


AVER- 
AGE 
HORSE 
POWER 
TRANS- 
MITTED 


AVER- 
AGE 
SPEED 

OF 
FRIC- 
TION 
WHEEL- 
R.P.M. 


AVER- 
AGE 

MILES 
PER 

HOUR 

(gear 

REDUC- 
TION 
4.5 TO 
I) 


I 


7.72 

5-97 
3-97 
1.97 


12.3 

13 -I 
14.0 

3-4 


1,440 

1,100 

700 

310 


305 
23.2 

14-7 
6.6 


6 


7.68 
5-93 

3-93 
1-93 


12.7 
12.3 

12.5 
6.1 


1,400 

1,120 

720 

370 


29.7 

23-7 

152 

7.8 


2 


7.72 
5-97 
3-97 
I 97 


13-6 

13 -I 
13-6 

3-7 


1,370 

1,065 

630 

340 


29.0 

22 .6 

133 

7-3 


7 


7.84 
6.09 
4.09 
2 .09 


10. 7 

10.4 

7.8 

4.1 


1,450 
1,180 

770 
425 


30.7 

25.0 

16.3 

8.9 


3 


7.72 
5-97 
3-97 
1.97 


13.0 

13.0 

9.8 

3-9 


1,380 
1,090 

740 
325 


29.4 

23.1 

15-6 

7.0 


8 


7.84 
6.09 
4.09 
2 .09 


12 . 2 
9.0 

8.8 
3-5 


1,340 

1,035 
650 
360 


28.3 
21 .9 

13 -7 
7.6 


4 


7.82 
6.06 
4.06 
2.06 


133 
12.7 

II-3 
5-5 


1,420 
1,125 

720 

340 


30.1 

23.8 

152 

7-3 


9 


7.78 
6.03 
4 03 
2.03 


6.3 

5-3 
3-2 


1,450 

1,115 
780 


31.0 

23-5 
16.5 


5 


7-75 
6.00 
4.00 
2 .00 


13.2 

10. 2 
5-4 


1,340 
1,050 

690 

340 


28.2 

22. 2 

147 

7-3 


10 


7.72 
5-97 
3 97 
1.97 


II .0 

8.5 
7.0 

3-5 


1,375 
1,100 

750 
300 


29. 1 

23.2 

158 

6.4 



POWER TRANSMITTING EFFICIENCY 

In finding the actual transmission efficiency between the driving disk and 
the driven friction v^heel it v^as found necessary to make proper allov^^ance for 
the mechanical losses in the driving end of the apparatus. These v^ere accu- 
rately determined by noting the amount of horse-pov^er required to run the 
disk idly at all the speeds desired. The percentage transmission efficiency 
therefore resolved itself into the simple formula: 



where 



E = 

E = 
Pi = 
Pd = 

Po = 

K = 



{Ft - Pd)K 

Po 



X 100 



the percentage efficiency. 

total horse-power. 

horse-power required to run disk idly. 

horse-power transmitted. 

constant for motor efficiency. 



FRICTION GEARS 293 

In Table 29 are given the actual percentage efficiency values. From a 
study of these it will be seen that in the case of the tarred paper Fiber No. i 
the maximum transmission efficiency was reached in the four positions at a 
contact pressure of 350-400 pounds (at 230-260 pounds per inch of working 
face). With the middle of the fiber at about 7.75 inches from the center of 
the disk, the efficiency rose steadily from 90 per cent, at a pressure of 250 
pounds to 95.5 per cent, at a pressure of 400 pounds every increase in pres- 
sure from there on tending to produce a decrease in the actual transmission 
efficiency, so that at 1,050 pounds it was but 82 per cent. This decrease in 
efficiency with each additional increase in pressure beyond that required to 
produce maximum results was true not only of this fiber and in this position 
with relation to the disk but also of the remaining fibers and positions. As 
Fiber No. i was brought in toward the center of the disk, the efficiency 
values decreased, giving peak values of 92 per cent., 90.6 per cent, and 69.9 
per cent, respectively. The per cent, slip increased from 3 per cent, at the 
7.75-inch or high speed position to about 20 per cent, at the 2-inch position. 
In the case of Fiber No. 2, the pressures required to produce maximum 
efficiency results are somewhat less than those in Fiber No. i. It must be 
noted that here the efficiency values themselves are all decidedly smaller. 
The slip is about twice as great. 

Except for the particularly great pressure which was required to secure 
the maximum value of 87.1 per cent, in the 4-inch position, the combination 
of brake lining and copper mesh Fiber No. 3 showed rather average results. 
The falling off in efficiency at high speed from the maximum at a pressure of 
350 pounds to that at 1,050 pounds was very gradual, being but 6 per cent. 
The slip at normal maximum running pressure in high speed position was 7 
per cent. 

Fiber No. 4 had a narrower working face than most of the other fibers. 
This may account for the high maximum efficiency value of 96.5 per cent, 
which occurred at 375 pounds and at 6 per cent, slip in high speed. As will 
be observed from Table 29, the results in the 6-inch position are almost as 
great as those in the 7.75-inch position. In low speed the pressure required 
to transmit the maximum horse-power appears to be 50-100 pounds higher 
than in high speed. This fiber gave smoother results than any of the 
others tested. 

Both Fibers Nos. 5 and 6 appear to have been entirely too soft to with- 
stand the rigorous testing to which they were submitted. In Fiber No. 6 
the results were extremely erratic. In Fiber No. 5 slipping was considerably 
greater than 20 per cent, at pressures up to 375 pounds in the high-speed 
position and at pressures up to 500 pounds in the low-speed position. The 
corresponding efficiency results were therefore very low. For Fiber No. 6 
the efficiency results proved to be far worse than those of any other mate- 
rial. In all four positions the maximum seems to have been reached at 350 
pounds pressure. From there on, each increase in pressure produced a very 



294 AMERICAN MACHINIST GEAR BOOK 

great diminution in transmission. Furthermore, the maximum efficiency- 
results themselves were extremely low. On account of the rapid disinte- 
gration of the material, it was impossible to secure other than the most 
unreliable speed readings of the friction wheel. The values for percentage 
slip are therefore omitted from Table 30 for all except the high-speed 
position. 

Fiber No. 7, also narrow in working face, gave efficiency values of 97.5 
per cent, at high speed and 95 per cent, at low speed (4 inches) both at a pres- 
sure of 300 pounds. In this fiber, as in Fiber No. 2, the pressure required to 
produce the maximum figure was very low. Slipping was almost negligible, 
increasing from 2 per cent, in the high-speed position to a comparatively 
small value of 9.6 per cent, in the 2-inch position. 

Fiber No. 8 behaved almost as uniformly as did the one immediately 
preceding it. The slipping was far in excess of the latter in all positions; 
the pressures required to produce the maximum transmission efficiency were 
greater; the percentage values were smaller but the actual decrease in effi- 
ciency as the fiber was brought in contact with the disk at points closer to the 
center was as uniform as it was in Fiber No. 7. This fiber showed maximum 
efficiencies of 95 per cent., 92.4 per cent., 91.8 per cent, and 80 per cent., 
respectively, for the four positions at 425 pounds pressure. The decrease in 
efficiency with increase in pressure was not in any way very pronounced. 

On account of the tendency for the oil to ooze out to the surface, Fiber No. 
9 presented results which were very inconsistent. The maximum trans- 
mission efficiency of 90 per cent., a comparatively low figure in the high-speed 
position, was attained at 300-350 pounds pressure. The decrease in effi- 
ciency from this pressure to that at i ,050 pounds was also greater than that 
shown by any of the fibers previously discussed, with the possible exception 
of No. 6. The values shown in Table 29 cannot be taken as a criterion of 
the way this fiber would behave under similar repeated tests. In fact, the 
high pressures in runs at both high and low speed forced the oil to the surface 
and caused the lubrication of the working face at 2 inches from the center of 
the disk the friction wheel slipped as much as 70-95 per cent. 

The rawhide Fiber No. 10, not unlike the other soft friction materials, 
gave very low maximum efficiency, results ranging from 87.8 per cent, in 
high speed to 70 per cent, in the lowest speed, at 350 pounds pressure. The 
drop in efficiency with increase in pressure also appears to be small. The slip, 
averaging 6 per cent, in high speed, does not seem to be greater than that 
shown by any of the treated paper or cotton fibers with the possible exception 
of the tarred Fiber No. i. 

A study of all of the above information does not bring out an absolute 
correlation between the amount of slipping and the degree of transmission 
efficiency. The increase in the latter does not appear to be entirely com- 
mensurate with the decrease in the former. Furthermore, it does not neces- 
sarily follow that, in the case of any two fibers, the one possessing the greater 



FRICTION GEARS 295 

transmission efficiency is submitted to less slipping. For instance, at a 
pressure of 375 pounds in high speed Fiber No. i shows an efficiency of 95 per 
cent, and a slip of 3.5 per cent., while Fiber No. 4 shows an efficiency of 96.5 
per cent, and a slip of 6 per cent. It is quite apparent, therefore, that factors 
other than the degree of slipping influence the transmission efficiency of any 
friction fiber. 

PHYSICAL PROPERTIES 

The length of time of operating each fiber and the pressures to which they 
were subjected were approximately the same throughout the test. In judg- 
ing the durability of the materials, therefore, no undue stress can be laid 
on a particular condition. The fibers were closely observed for physical 
appearance before, during and after operation, for temperature and noise. 

Fiber No. i. Was subjected to no glazing or burnishing action under 
high slipping such as is usually encountered at contact pressures up to 300 
pounds. As the pressures were increased from 450 pounds up to 1,050 
pounds, especially in the low-speed positions, the fiber grew hotter. After 
the check run in the 2-inch position it gave slight evidence of burning. 

Fiber No. 2. Showed a decided tendency to burning at both high slipping 
and with increase in pressure above that required for normal operation. It 
was noticed that after cooling, the fiber had become somewhat harder than it 
was originally and the radial segments had twisted out of their original posi- 
tion, the degree of twist increasing as the fiber was brought closer to the 
center of the disk. 

Fiber No. 3. In spite of the great amount of slipping, especially when 
tested in the 4-inch position, this fiber showed but little tendency to 
become hot. The copper mesh in the fiber had a decided cutting effect upon 
the disk. This fiber, like all of the remaining hard fibers, produced consider- 
able noise in operation. 

Fiber No. 4. At a pressure of 300 pounds the slip was so great as to produce 
rapid burning and a resultant odor of formalin. However, when this fiber 
not permitted to slip excessively, it does not heat up like any of the paper 
material, even after the application of the higher pressures of contact. 
Because of the failure on the part of the manufacturer to provide a width for 
the total thickness of the filler in excess of that of the working face and also 
bevelled edges on either side as retaining walls, this fiber showed a tendency to 
break down over the supporting flanges. 

Fiber No. 5. Burned quickly when subjected to slip even for a very 
short period of time. In the course of operation, especially in the low-speed 
positions, the tendency to smoke was pronounced. At the end of the test 
the material showed a deep groove running through the middle of the ring, 
caused presumably by charring. 



296 AMERICAN MACHINIST GEAR BOOK 

Fiber No. 6. Proved to be entirely too soft to withstand the higher 
pressures of contact. It also suffered from a poor, uneven finish given the 
working face by the manufacturer. In order to prolong the life of the fiber, 
every precaution was taken to reduce slipping to a minimum. At pressures 
of 700 pounds in the high-speed position, the fiber began to smoulder. The 
speed run in the same position produced broken uneven surfaces. In the low 
check positions the effect of burning was so great as to cause almost complete 
disintegration, great tufts of the material being virtually torn out of the filler. 

Fiber No. 7. Showed slight tendency to slip in the short runs at low 
pressure and therefore suffered but little of the effects attending such a con- 
dition. There was also almost a negligible amount of heating at the high 
contact pressures. However, the fiber was extremely hard, glazed somewhat, 
and produced a peculiar siren-like noise when operated in the high-speed 
positions. 

Fiber No. 8. Appeared in many respects to behave like Fiber No. 7. 
The effect of heating was very slight. It showed a glazed surface and pro- 
duced rather more noise in operation than any of the fibers except No. 3 and 
No. 7. 

Fiber No. 9. Is not believed to have been constructed with due regard 
to the severe conditions under which variable speed disk drives operate. 
In the runs in both high- and low-speed positions, the oil in the filler constantly 
came out to the surface, lubricating the disk and working face, causing con- 
siderable slipping and the consequent generation of great heat. 

Fiber No. 10. Was one of the softest fibers of the ten. Under the ordi- 
nary operating conditions it behaved in many respects like Fiber No. 5, 
becoming just as hot. It expanded considerably, curled over the retaining 
wall and caused an unevenness and increase in width of the working face. 

RESULTS OF ENDURANCE TESTS 

In order to secure a better idea of the durability of the materials, Fibers 
Nos. I, 2, 4, 7, and 8, considered at this time to be the best of the ten fibers 
under investigation, were subjected to fairly strenuous runs in the position 2 
inches from the center of the disk. In this position it was believed the great- 
est amount of slipping and twisting would cause the most rapid breaking 
down of the materials. The motor speed averaged 1,650-1,725 r. p. m., 
the speed of the friction wheel at the supposed zero slip was calculated to be 
about 390-405 r. p. m. The pressure of contact used was 425 pounds, 
which it was previously found had given the average maximum efficiency 
values in the 2-inch position. Results of these runs are submitted in Table 
30 below: 



FRICTION GEARS 



297 



TABLE 30 



FIBER 
NO. 



TOTAL 

NO. 

HOURS 

OF 

RUN 



TOTAL NO. 
REVOLU- 
TIONS MADE 
BY FRICTION 
WHEEL 



TOTAL NO. 
OF MILES 

(internal 
gear ratio 

4.5 TO l) 



AVERAGE 

R.P.M. 

OF 

FRICTION 

WHEEL 



AVERAGE 




TRANS- 


AVERAGE 


MISSION 


SLIP, 


EFFICIENCY, 


PER 


PER CENT. 


CENT. 



AVERAGE 
REDUCTIOn 

IN O.D. 

OF WHEEL, 

INCH 



28.0 
32.1 
27 . 2 
31.6 
32.2 



466,422 
513,586 
485,107 

339,122 
578,422 



164.4 
181. 6 
170.9 

iiQ-S 
204.5 



277 


79-4 


28.7 


267 


70.7 


33-3 


298 


81. 5 


23 -7 


177 


56.0 


55-3 


300 


76.3 


26.3 



0-035 

0.030 

0.085 
0.050 



The run of the tarred Fiber No. i was not continuous. It extended over 
four periods each of approximately seven hours' duration. Fluctuations in 
transmission efficiency and slip, evident in some of the other materials sub- 
mitted to this endurance test, were not quite so apparent in this fiber. 
Starting with the friction members cold, the efficiency value was comparatively 
low. However, as the fiber warmed up gradually the percentage efficiency in- 
creased so that for the last average hour of operation in the period the efficiency 
was at a maximum. The amount of slip increased with increase in efficiency, 
thereby producing a decrease in the relative speed of the friction wheel. 
This phenomenon also held true in the case of the remaining fibers so tested. 
Examination of the fiber at the end of twenty-eight hours showed almost no 
perceptible reduction in its outer diameter, an increase of about 4 per cent, 
in width of working face and gave evidence of some burning on the inner edge 
of the face. Taking into consideration the fact that it was operated at a 
position on the disk much closer to the center than is ever approached in 
ordinary practice, the fiber proved to be of extraordinarily good material and 
showed very good efficiency results. 

A careful study of the table shows readily that the compressed cotton 
duck Fiber No. 2 fell somewhat below fiber No. i in its transmission efficiency. 
Although operated under conditions almost identical with those of the 
material just described, this fiber showed decidedly worse results in every 
respect; a greater average slip, a more pronounced tendency to slip with each 
continuous run and the consequent decrease in the speed, and lower efficiency 
throughout. 

The fiber remained unchanged in width of working face. The changes 
in outer diameter were as follows: Before test — outer edge, 16.687 inches; 
inner edge, 16.687 inches; after test — outer edge, 16.672 inches; inner edge 
16.703 inches It is to be noted that where the outer edge of the fiber was worn 
away about 0.015 inch, the inner edge increased in diameter by fully 0.016 
inch. This can be attributed to the fact that the twisting action on the fiber 
threw the radial pieces out of the lateral position at a point about J^ inch from 
the inner edge, the twist increasing at points closer to the inner edge. No 
doubt such action served to relieve the compression under which the cotton 



298 AMERICAN MACHINIST GEAR BOOK 

duck pieces had been laid into the wheel and to produce the consequent 
increase in outer diameter. 

The ''micarta" Fiber No. 4 appeared to be the best of all the fibers 
submitted to the endurance test, showing a fair uniformity in transmission 
efficiency and degree of slip in the four seven-hour runs. The increase in 
efficiency from the first to the seventh hour averaged only 5 per cent. As in 
the case of Fiber No. i, the slip increased as the day's run progressed. How- 
ever, the average efficiency and average slip proved to be the highest and 
lowest, respectively, of the five fibers tested. Because of poor design, 
especially lack of the bevelled edges' previously discussed, this material 
became frayed somewhat. The slight burning action decreased the outer 
diameter uniformly over the whole working face by 0.03 inch. The relatively 
favorable behavior under these rather arduous conditions lead to the belief 
that this fiber would prove to stand up very well in every respect in practice. 

Fiber No. 7 shattered every favorable impression previously created in 
regard to good transmission efficiency and durability. In three runs, each of 
nine hours, the efficiency fell from a value of about 85 per cent, for the first 
hour to about 25 per cent, for the ninth hour; the slip increased gradually 
from about 25 per cent, to about 85 per cent.; the speed of the friction wheel 
decreased from approximately 300 r. p. m. to 50 r. p. m. On one occasion the 
test was interrupted after a continuous run of four hours for several minutes 
and then resumed. The result of this was to bring the efficiency and speed 
back to the maximum point. Atmospheric temperature changes such as were 
produced by opening a window near the apparatus caused decided reductions 
in the speed and transmission efficiency. A feature of this fiber was the 
increasing noise in the course of operation. It commenced as a dull, heavy 
sound, developed into a sort of crunching noise and finally produced a rumble 
not unlike that of a heavy stone-crushing mill. After completion of the 
test, accurate calipering of the working face revealed a decrease in diameter 
of 0.07 inch on the outer edge and o.io inch on the inner edge. 

Fiber No. 8 showed results favorably comparable in every respect with 
those of Fiber No. i. The variation in transmission efiiciency was almost 
negligible, being for the most part very close to the average indicated in 
Table 30. The percentage slip, almost constant throughout, was less than 
that of the tarred fiber and consequently the decrease in speed in the course 
of the eight- or nine-hour runs was very slight. Like Fiber No. i , this material 
was subjected to slight burning, especially on the inner edge of the working 
face. This was evidenced by the following diminutions in outer diameter: 
Before test — 17.00 inches; after test, outer edge — 16.96 inches; inner edge — 
16.94 inches. 

COEFFICIENT OF FRICTION 

For disk drives wherein the increase in the amount of horse-power trans- 
mitted varies with the increase in contact pressure, i.e., up to the pressure 



FRICTION GEARS 



299 



producing the maximum transmission efficiency, the value of the coefficient of 
friction is practically independent of the pressure of contact. Furthermore 
(as is claimed on authority), "excluding positions at the extreme center, it is 
independent of the position of the fiber wheel on the disk." For this reason 
and in order to adhere to a strict comparison of the coefficient values it was 
deemed advisable to base the calculations on the results secured from tests in 
the 4-inch position. It will be remembered that in this position every 
fiber was in contact with the aluminum alloy disk. The values for coefficient 
of friction are given for pressures which produced the maximum transmission 
efficiency. To derive the results tabulated below, the general formula for 
friction wheels was used: 

^.1^16 DN 



H.P. = 



12 



XPp 



33,000 



D 

N 
P 

P 



Mean diameter of friction wheel in inches. 
No. revolutions per minute. 
Pressure of contact over working face. 
Value of coefficient of friction. 









TABLE 


31 














Fiber No 


I 


2 


3 


4 


5 


6 


7 


8 


9 


10 


i 




Pressure of contact, pound 
Coefficient of friction 


3750 
0.368 


425.0 
0.362 


425.0 
0. 260 


425.0 
0.316 


475-0 
0.304 


350.0 
0.392 


300.0 
0. 276 


425.0 
0.250 


3750 
0.070 


37S-0 
0.218 



From the results given it is readily seen that the four highest values for 
the coefficient of friction are shown by fibers Nos. i, 2, 4 and 6. In the case 
of fiber No. i the value submitted is almost identical with the manufacturer's 
given safe value of 0.364 for tarred fiber and zinc alloy and approaches 
closely the value of 0.390 for tarred fiber and aluminum alloy. For fibers 
2 and 4 (the most efficient fiber) the value is not considered low. Fiber No. 
6, being fairly soft material, presented the highest coefficient of friction. 
Fibers Nos. 3, 7 and 8, all very hard materials, appear to have suffered from 
the fact that hardening a material usually reduces the coefficient value. The 
low results in fiber No. 9 can be attributed to the oil at the surface. That 
fiber No. 10 possesses a low value is not surprising for the low coefficients 
of friction of leather against metal are well known. 



DISCUSSION OF EXPERIMENTAL RESULTS 

Two elements appear to influence the suitability of friction fibers with 
regard to transmission capacity and durability under the conditions of. 
service; the pressure of contact and the value of the coefficient of friction 



300 AMERICAN MACHINIST GEAR BOOK 

between disk and friction wheel. Since the power capable of being trans- 
mitted varies with the pressure of contact it is believed that a fiber of high 
compression strength will give longer service. However, as the friction 
material is increased in hardness to withstand the greater pressures, the high 
value of the coefficient of friction decreases. Still another limiting factor, as 
set forth in this investigation, is the efficiency of transmission. The actual 
decrease in efficiency value with additional pressures beyond that intended to 
produce the maximum is now well known. 

From one point of view it might be advisable that the fiber transmit the 
power with maximum efficiency at a low contact pressure, thereby saving the 
material from the effects of high compression. The heavy requirements of 
disk friction drives are known, however; the frequent applications of high 
contact pressures under all sorts of conditions even for short periods of time 
necessitate the material being sufficiently hard. That the transmission 
efficiency at those pressures will be less and that it will be increasingly difficult 
to secure a high value for the coefficient of friction is also fairly certain; but 
if a fiber can be secured wherein it would be possible to retain a good coeffi- 
cient and to keep the decrease in transmission efficiency at a minimum, the 
elimination of present difficulties will have been accomplished in a great 
measure. 

A serious fault of the present friction fillers is the insufficiently developed 
or, as appeared from one or two of the fibers tested, the absence of pressure 
withstanding quality. Even such fairly hard materials as the tarred fiber 
No. I or the "micarta" fiber No. 4 suffered from the highcontact pressure. Just 
what takes place when a fiber is subjected to high pressure is best described 
by W. D. Hamerstadt in a recent paper entitled, "History and Development 
of Friction Drive." He states that "when a pressure is applied by a metal 
against a fiber wheel, there results a certain flexing or compression of the 
fiber material. This results in the disturbance of the structure of the fiber 
in the relation of adjoining individual fibers in the mass. Should this pres- 
sure be great enough, it is readily conceivable that the bond or interlocking 
of certain fibers with the adjoining ones becomes broken. This results in a 
sheering action through the fiber as encountered in cast iron, steel, wood, 
stone, or cement, in a compression test. Having once broken the original 
bond or structure, it now becomes much easier for the fiber to compress with 
subsequent applications of pressure as the wheels pass in contact at high 
speeds, this working of one fiber on another soon results in generating very high 
heat. This heat is the damaging factor, as it very soon dries out the fiber — 
makes it brashy like charred or burned leather and the further destruction of 
the material is very much hastened." This action was ideally set forth in 
the applications of high pressures to fiber No 6. The constant kneading of 
one fiber over the other produced the high heat which caused the rapid dis- 
integration of the material. Fibers No. 5 and No, 10, insufficiently hard, 



FRICTION GEARS 301 

likewise suffered from the same action. In cases where the fiber is not 
brought in contact with the disk to prevent excessive shpping, the heating 
effect is decidedly more rapid. Here it does not suffer from the shearing 
action and working of one fiber upon another with the consequent drying of 
the material. The action is, on the contrary, that produced by sliding fric- 
tion. The constant slipping produces a high heat and charring on the 
surface which gradually works its way inward and causes brittleness and 
disintegration. 

In the light of all these generalizations, it is possible to classify our mate- 
rials into two groups: (a) Hard fibers; (b) soft fibers. In the first classifica- 
tion can be placed fibers Nos. 3,7,8 and 9. A careful examination of Table 31 
will shov/ that the values of coefficient of friction for these fibers are all very 
low. Fiber No. 4 in the true sense is also a hard material but its coefficient 
of friction is fairly high and its general behavior in every way is such as to 
make its classification advisable under that of soft fibers. From Table 28 it 
will be readily seen that the horse-power transmitting capacity of all these 
fibers, especially in the low-speed positions, is lower than that for the soft 
fibers. As regards transmission efficiency, these fibers gave good results in 
the short tests, especially in the high speed positions; but the great amount of 
slip and the general disappointing behavior of Fiber No. 9 throughout and of 
Fiber No. 7 in the endurance test is now apparent. It is believed that any of 
these fibers, if submitted to actual operation, over long periods of time would 
give poor efficiency in the long run, produce great slipping and because of 
the low coefficient of friction transmit little horse-power. 

Of the so-called soft fibers, Nos. 5, 6 and 10 must be eliminated from 
favorable consideration, the first two because of very poor pressure with- 
standing qualities and the last because of its general softness and the very low 
coefficient of friction. These three fibers have not only shown a low value for 
maximum transmission efficiency but also decided falling off in percentage 
with the increasing pressures throughout all four positions on the disk. Of 
the three remaining fibers. No. 2 has given the lowest efficiency results, 
standing out particularly as a poor transmission medium at high contact 
pressures. In spite of the tendency on the part of the radial pieces of cotton 
duck to twist out of a lateral position, this fiber has shown good wearing 
qualities. The tarred paper fiber No. i and the '^Micarta" fiber No. 4 
therefore stand out as the foremost materials in every respect. In the 
endurance test the fibers stood up equally well. Both fibers have shown 
equally good horse-power transmitting capacity. Fiber No. 4 showed the 
higher maximum transmission efficiency throughout and a decidedly lower 
decrease in efficiency as the pressures increased beyond those which gave the 
maximum value in each position. On the other hand, this fiber showed a 
greater percentage slip in the high speed positions and possessed a lower 
coefficient of friction, 



302 AMERICAN MACHINIST GEAR BOOK 

From a consideration of all the results submitted it would seem advisable 
to concentrate further investigation as regards efficiency and durability 
under other conditions of operation and to make further developments and 
favorable changes in composition and design upon fibers Nos. 1,2, and 4. 
Such a course, it is believed, is preferable to overcoming the many weaknesses 
presented by the remaining fibers." 



SECTION XIV 

Special Bevel Gears 

The unprecedented demand for high grade bevel gears and ring gears 
which can be produced in large quantities economically for automobile 
drives has led to the development of certain distinctive types of bevels, which 
can very properly be classified as special gears. For the most part the tooth 
forms of these gears differ radically from that of the customarily employed 
octoid system with radial teeth and their production entails operations 
previously foreign to gear manufacture. 

SPIRAL TYPE BEVEL GEARS 

The Gleason Works have developed one of the most distinctive of these 
special gears (Fig. 147) in which the teeth are arranged, not in a spiral manner 




-H k- 



FIG. 247. DIAGRAM OF SPIRAL TYPE BEVEL GEARS. 



as the designation of the gear would imply, but in the form of circular arcs 
which if prolonged would intersect at the cone center of the gear. That is, 
the various curved tooth elements converge and would all intersect at the 
cone center. Another pecuHarity of the gear teeth is that, though they are 

303 



304 



AMERICAN MACHINIST GEAR BOOK 



generated to true octoid form by means of rotary cutters ground to rack form, 
the pressure angle (contact slope) is slightly different for either side of the 
teeth. This is due to the fact that the radii of the two tooth profiles, the 
concave and the convex, differ by the thickness of the rotary cutting tool, 
opposite sides of the cutter being employed to cut the two sides of the gear 
teeth, the outer edge of the cutter forming the concave sides of the gear teeth 
and the inner edge the convex sides. 

Another characteristic of spiral type bevel gears is the peculiar thrust 
developed by the gears in action, diagrammatically depicted in Fig. 248. 
The transmitted load is naturally normal to radial elements of the gear, while 
the total tooth load is normal to the tooth. Considering the tooth pressure 
concentrated at the center point of the tooth, arrow A (Fig. 248), either 



<u-ffet-/?ccf/^^___ 




FIG. 248. DIAGRAM OF RIGHT-HAND SPIRAL ILLUSTRATING PRESSURES IN FORWARD AND 

REVERSE ROTATION. 



above or below the center line of the tooth, depending upon the direction of 
rotation, represents the total tooth load; arrow B, the transmitted load; and 
arrow C, the balancing load, represents the thrust either in toward the apex 
center of the gear, or in the opposite direction, depending upon the direction 
of rotation. In addition to this spiral angle thrust is the ordinary pressure 
angle thrust common to all types of bevel gears. For rotation in one direc- 
tion, this pressure angle thrust augments the spiral angle thrust, while for 
rotation in the reverse direction, the spiral angle thrust is toward the center 
of the gear and the pressure angle thrust acts against it, so the resulting 
thrust is the difference between the two forces. 

The direction of rotation of a pair of spiral type bevel gears is customarily 
referred to that of the pinion. Viewing the pinion from the rear, rotation in 



SPECIAL BEVEL GEARS 



305 



a clockwise direction is designated as forward and in an anti-clockwise direc- 
tion as reverse. In the case of the gear, the rotation is, of course, opposite 
and a positive spiral angle is taken as one in which the advancing profile 
planes of the teeth are convex in forward rotation (see Fig. 248). 

The pressure angle thrust which either increases or reduces the spiral 
angle thrust really depends upon several conditions, such as the amount of 
power transmitted, pitch diameter and pitch angle of the pinion, revolutions 
of the gear combination and the spiral angle, but, for all practical purposes, 



80 



o 
o 



T5 



IS 
TO 
65 



^ 60 



o 

•+- 

c 



.•i 



55 
50 

45 
40 
55 
30 



CT) 25 







20 



^f'Thwsf ihdicahs Pressure awai/ frvm the Gear _ 
— " Thwsi indicai-es Pull in toward the com center 

^i I I I— 



IZZ 



I I I I 



20 



40 



CHART 16. 



25 30 35 

Spiral An^Ie^de^rees 

VALUE AND DIRECTION OF THRUST IN SPIRAL TYPE BEVEL GEARS. 



may be reduced to a mean percentage of the load transmitted, or of the spiral 
thrust which is a definite proportion of the transmitted load for any given 
angle of spiral. Chart 16 depicts the average total load thrust for spiral 
angles of from 20 to 40 degrees in percentage of transmitted load, the plus 
values indicating pressures away from the gear and the minus values pressures 
toward the gear apex. 

The increased load thrown on the teeth by reason of their spiral arrange- 
ment—the transmitted load divided by the cosine of the spiral angle— neces- 
sitates that the load be distributed over several teeth, so if the same general 
tooth form as is employed for straight tooth bevels is to be employed there 
is a definite relation between the lead, or spiral angle, and the circular pitch 
of the gears. The advisable relation between the spiral angle and the circu- 



3o6 



AMERICAN MACHINIST GEAR BOOK 



lar pitch is shown on Chart 1 7 and also the total tooth load in percentage of 
transmitted load for the various spiral angles between 20 and 40 degrees, the 
advisable lead being made proportional to the total tooth load plus a safety 
allowance of 10 per cent. 

The pitch of the gear also affects the question of advisable spiral angle, 
as the finer the pitch the smaller can be the spiral angle for a given radius of 
tooth curve. It is desirable, furthermore, to employ cutters of the same diam- 
eter for the various pitches and sizes of gears which may be manufactured 
so the hypothesis may be made that the advisable spiral angle varies directly 



S 



-T3 



0} 



o 

s 



1 



it 



o 



I 



I'+O 








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/ 
















































































/' 
















































































/^ 






140 










































































/" 














































































y 


























To fa'/ Tooth load in PerCenfx /. / 




































T 


z. 




1 
















- Leaa= 
1 r , 




Ci'rcu/ar Pitch , , 






— 


— 




— 




— 


' 


— 


-— 


— 




y 






— 


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— 


— 


135 
































































^' 












































































^ 


^ 














































































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^ 














































































, 


y 




























130 




















































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i^ 
















































^ 






























y 


















































X 






























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^ 






























, 


^ 
















































^ 






























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^ 










125 




































^ 
































<^ 














































































^ 














































^ 


































y 












































^ 




































^ 










































r^ 




































^ 
























120 


















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^ 




































^ 


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.— - 


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rn'- 
































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115 


.Li 


-CI 


































^ 
























































































































































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^ 














































































.^ 
























































no 




















^ 


^ 












































































^ 














































































n 


r,n 
































/ _ 


'it ' ' -I, 1 
























— -r^/- 


^1^ 


L- 


















Tot'^l Thnfh 1 n^rl- 






















"Y^ 


} '^" 




























( 


Tosine ofSoiralAnale 
















105 
















































































































































































































































































































































































































ino 








_ 


_ 









_ 


_ 


__ 




























_ 




_ 




_ 


_ 


^ 




























20 



25 



50 



Spiral Angle, degrees 



35 



40 



CHART 17. 



EFFECT OF SPIRAL ANGLE ON TOOTH PRESSURE AND MINIMUM ADVISABLE LEAD 

SPIRAL TYPE BEVEL GEARS. 



with the diametral pitch of the gear. Based on such relation. Chart 1 8 depicts 
good practice. 

As the amount of lead for a given spiral angle and constant cutter radius 
depends upon the face of the gear, there is from the point of view of practical 
design an advisable minimum face for a given angle of spiral. A face any 
wider than such minimum permits and generally calls for a reduction in angle 
of spiral, if the cutter radius is not increased, for unless this is done too much 
importance is placed upon the perfection of workmanship in cutting the 
gears. The wider face secures no other advantage than an increase in power 
capacity of gears and this in the case of spiral type bevel gears is better 
regulated by the angle of spiral than it is by the face of the gear. However, 



SPECIAL BEVEL GEARS 



307 



the face of the gear should be sufficient to enable the advisable lead to be 
secured for the given angle of spiral with fixed radius of cutter. 



\\J <n- 


-t 


.., 


: ::'s 







N 


y 


.., 




+__ 5s---: 




- TP* 4-t- 





II ^. ^ 




V _L, 




'^ N 




_^% 




_..^^i._.4+ 


1 


+ .___!s 14 




"^,t ^ 




^ -.. S 1 ^ 


-^ 6 


^..._!v 


p "^ 


_ -^5. 


-+- 


+_ _^^s---++^ 


D- 


__._^^, 




^....!> 1 


° 


^:. .s, . 


C 




"^^ 


■^^ 


"*v 


t ^ 


"N- , 





^^ 


Ci. 




V 


^ 


''y^ 




■^ V 




-^i-- -^ 





^ ^i 




^.._i. 




X"s -1— 




±:'!s:::::^ 


1 










T 





*^^^S 



15 



20 



40 



45 



25 30 35 

Advisable Spiral Angle^d^rees 

CHART 18. DIAMETRAL PITCH AND ADVISABLE SPIRAL ANGLE SPIRAL TYPE BEVEL GEARS. 




FIG. 249. RELATION BETWEEN FACE AND SPIRAL ANGLE SPIRAL TYPE BEVEL GEARS. 

Fig. 249 illustrates the relation between the face of the gear and the 
spiral angle. The angle efg is equal to the spiral angle and the dimension 



3o8 AMERICAN MACHINIST GEAR BOOK 

V is equal to the face of the gear multiplied by the tangent of the angle of 
spiral and is always slightly less than the actual lead. Consequently, if the 
face is made equal to the lead divided by the tangent of the spiral angle it 
will be slightly in excess of that required to secure the lead. This slight 
excess width can be ignored and the formula for minimum advisable face of 
gear becomes: 

^_ L 
tan. A 
Where, L = lead, A = spiral angle, h — face. 
Referring to Fig. 249, 

L' = h' tan. A <L (slightly less) 



tan. A tan. A 



TOOTH PROPORTIONS 



The octoid form of tooth with a mean pressure angle of i^^^i degrees has 
been adopted as the standard for spiral type bevel gears. The blades of the 
cutter employed for forming the teeth are ground to rack form, so that the 
cutter and gear blank rolling together with a generating motion produce a 
true octoid tooth, but the curved arrangement of teeth necessitates a slight 
modification of the standard 143^^-degree octoid tooth, the pressure angle on 
the convex side of the teeth being made slightly greater than 143^^ degrees and 
on the concave side slightly less. The increase and decrease are equal, 
however, so that the sum of the two pressure angles is always 29 degrees. 

As the spiral type of bevel gear permits and is usually employed in 
greater speed ratios than can be realized with straight tooth bevels, there is 
danger of undercutting the pinion teeth if the addendum and dedendum 
dimensions of the gears are made equal, so the standard adopted by the Glea- 
son Works is to make the pitch depth of the pinion teeth seven-tenths of the 
total working depth of the tooth and the pitch depth of the gear, conse- 
quently, three-tenths of the working depth. This modification affects the 
face and cutting angles of the gears and the outside diameters of the gear 
blanks. 

MACHINING OPERATIONS 

In machining the teeth of spiral type bevel gears, two cutters are 
employed. The teeth are first blocked out with a roughing cutter and then a 
finishing cutter is employed to finish first one side of the teeth and then the 
other, the same finishing cutter being used to finish both sides of the teeth, 
but on slightly different cutter settings. These cutter settings must be accu- 
rately made and two methods of doing this have evolved — the "formula" 
method and the "layout" method. 

The horizontal and vertical settings of the roughing cutter center for 



SPECIAL BEVEL GEARS 



309 



Horizoni-cil Seffing 
< RsinA -X- r > 



, y' Q-Cuffer Cenh?r 



J> / /Angle 

^ ' ' of Spiral 



.0 




FIG. 250. SETTINGS FOR ROUGHING CUT — SPIRAL TYPE BEVEL GEARS. 




_ . . Piich Line -BoHom Side ofToofh 

^^"^^ "CenlerLine ofToo-fh 

FIG. 251. SETTINGS FOR FINISHING CUTS — SPIRAL TYPE BEVEL GEARS. 



3IO AMERICAN MACHINIST GEAR BOOK 

right-hand spiral gears are shown in Fig. 250 and those for the finishing cuts 
in Fig. 251. The equations employed for computing the various dimensions 

according to the "formula" method follow and a brief explanation of their 
derivation. 

NOTATION FOR SPIRAL TYPE BEVEL GEARS 

DESIGN AND GENERAL GEAR PINION 

Diametral pitch p p 

Circular pitch p' p' 

Number of teeth n n' 

Spiral angle A A 

Lead L L 

Face (actual) h b 

(minimum) b' b' 

Depth of tooth W W 

Addendum s s' 

Dedendum u u' 

Clearance / / 

Circular thickness of tooth (outer pitch circumference) cl ct' 

Center angle E E' 

Pitch diameter d d' 

Cone distance a a 

Angle increment / /' 

Angle decrement K K' 

Face angle F F' 

Cutting angle C C 

Diameter increment V V 

Outside diameter D D' 

SHOP 

Diameter of cutter D" D" 

Radius of cutter (mean) R R 

Inner cutter angle B B 

Outer cutter angle T T 

Thickness of cutter at apex t" t" 

Horizontal setting Y Y 

Vertical setting (roughing cut) X X 

Vertical setting (finishing cuts) 

Top side of tooth (concave) X' X' 

Bottom side of tooth (convex) X" X" 

FORMULAS FOR SPIRAL TYPE BEVEL GEARS 

Test Formula: 

b' = 7= 7 or, advisably Tan. A = -: Formula A. 

Tan. A b 

FORMULA 

Design Calculations: 

Tan. E = — i 

n^ 

E' = go - E i' 

d = — 2 

P 

d' = '^ 2' 

P 

W = — - (i4>^-deg. standard tooth) 3 

P 

pitch depth X 2 0.6 

s=- f = — 4 

P P 



SPECIAL BEVEL GEARS 311 

, _ pitch depth X 2 _ 1.4 , 

P P 

u = W' - s 5 

u' = W' - s' 5' 

ct = 0.5 p' — — (14^^-deg. standard tooth) 6 

P 

ct' = 0.5 p' -\ — ' (i43'^-deg. standard tooth) 6' 

P 
__ o-s d _ 0.5 d' 

Sin. E Sin. E' 

Tan. J =~ 8 

a 

Tan. / = - 8' 

a 

Tan. K =— g 

a 

Tan. K' = — g' 

a 

F = E + J 10 

F' = E' -{-r 10' 

C = E - K II 

a = E' - K' 11' 

V = s COS. E 12 

V = s' COS. E' 12' 

D = d-\- 2Y 13 

D' = d' ^ 2V' 13' 

NOTATIONS FOR SHOP CALCULATIONS 

GEAR PINION 

Outer cutting radius (concave side — mean value) R' R\ 

Inner cutting radius (convex side — mean value) R" Ri" 

Dedendum at mid-tooth u" U\' 

All other notation similar to that employed for design calculations. 

FORMULAS FOR SHOP USE 

FORMULA 

„ _ (g - 0.56)^ 

u = 1 

a 
^„ ^ (a - o.sbW J, 

a 
R' = R-\- o.sit" + 0.5172 w") II 

Ri' = R-\- o.5(r + 0.5172 ui") ir 

R" = R- o.sit" + 0.5172 u") Ill 

2?i" = R- o.5(r + 0.5172 u,") iir 

Y = a - (0.5 b + R sin.A) IV 

X = R COS. A V 

X' = ^ VI 

K 

x,' = ^ vr 

K 

X" = ^ VII 

X," =^ VIII, 



312 AMERICAN MACHINIST GEAR BOOK 

DERIVATION OF FOREGOING FORMULAS 

All formulas pertaining to tooth proportions, pitch diameters, center 
angles, etc., which are not affected by the pitch depth of gear or pinion are 
derived in a manner similar to that for equivalent formulas for straight tooth 
bevel gears. 

The addendum of the gear or pinion is found by dividing the proportional 
pitch depth by the diametral pitch. For gears in which the pitch depth 
equals 0.3 of the working depth of tooth, this is equal to 0.6 divided by the 
diametral pitch, and for pinions in which the pitch depth equals 0.7 of the 
working depth equals 1.4 divided by the diametral pitch. 

The dedendum is most readily figured by subtracting the addendum from 
the working depth of the tooth, which latter is the same as in the case of 
ordinary straight tooth bevel gears. 

The circular thickness of tooth on the pitch circumference of the gear is 
equal to half the circular pitch minus twice the product of the tangent of the 
pressure angle by the difference between the pitch depth and half the working 
depth of the tooth, divided by the diametral pitch. The circular thickness of 
the tooth on the pitch circumference of the pinion is equal to half the circular 
pitch plus this same amount. 

The cone distance of spiral type bevel gears is the same as that for ordi- 
nary bevels, but the angles increment and decrement are affected by the pitch 
depth of the gear and pinion, their tangents being respectively the addendums 
and the dedendums, divided by the cone distance. 

The face and cutting angles of the gears and pinions are simply affected 
by the change in angles increment and decrement, the formulas being similar 
to these for straight bevel gears; likewise, the diameter increment and the 
outside diameters of the gear and pinion. 

The finishing cutter radii must necessarily be proportioned to the bottom 
width of the tooth spaces, necessitating taking into consideration the relative 
position of the pitch line: i.e., pitch depth of gear and pinion. The outer 
cutting radius, that for the concave side of the tooth, is then equal to the mean 
cutter radius, plus one-half the apex thickness of the cutter, plus the deden- 
dum (at center point of tooth) times the tangent of the outside cutter angle. 
The inner cutting radius is equal to the mean cutter radius, minus one-half 
the apex thickness of the cutter, minus the product of the mean dedendum and 
the tangent of the inner cutter angle. 

The horizontal and vertical settings for both roughing and finishing cuts 
have already been explained, but it must always be borne in mind that the 
formulas presented for the finishing cut settings are based on the respective 
cutting radii normal to the pitch planes at the center point of the tooth pro- 
files, not at either the outer or inner pitch circumferences. The finishing 
cutter radius (for either side of the tooth) varies from outer to inner end of 
tooth, but by locating the vertical setting from the cutting radius at mid- 



SPECIAL BEVEL GEARS 



313 



tooth the angle decrement automatically causes the required variation in 
radius on the pitch plane and over the entire tooth profile. 



.o,^» 



10-ou 




























































































































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r3 14-30 





























































































































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Diamefrai Pi>ch 

CHART 19. 

The accepted standards for the finishing cutter angles, inner and outer, 
for different diametral pitches are given on Chart 19 and the details of the 




< 

<--■ 

-Diame-hr of CuHer, D' 



FIG. 252. DETAIL OF FINISHING CUTTER— SPIRAL TYPE BEVEL GEARS. 



finishing cutter are shown in Fig. 252. The mean radius of the cutter, R, 
equals one-half the diameter of the cutter, D" , while the cutter angles ^ and T 
are obtained from Chart 19 and t" is arbitrarily set customarilv at Kg inch. 



314 



AMERICAN MACHINIST GEAR BOOK 



THE "LAYOUT METHOD" 

This method of obtaining the vertical settings for the finishing cuts is 
based on a somewhat different theory than that governing the formula method, 
and, as its advocates claim greater precision results from its use, a brief 
description of this method is of interest. 

Referring to Fig. 253, a diagram of the gear is laid out on an enlarged 
scale in which the curve Im depicts the center line of the tooth — the curve 
being the arc of a circle having its center located by the horizontal and 




FIG. 253. 

vertical settings for the roughing cut and a radius equal to the mean radius of 
the cutter. For the top side of the tooth (concave), the points /' and m' are 
struck off from points I and m respectively on the radii to these points — i.e., 
on lines radiating from the cutter center to the points of intersection of the 
center line of the tooth with the inner and outer pitch circumferences — //' 
and mm' equaling the dedendum times the tangent of the pressure angle at 
the respective points, the ratio //': mm' equaling a: (a-b). With the 
points /' and 7n' as centers and a radius equal to the mean radius of the cutter 
plus one-half the thickness of the cutter blade at its apex arcs are struck off 
intersecting at Q'. 



SPECIAL BEVEL GEARS 



315 



For the bottom side of the tooth (convex), similar steps are taken, locating 
the point Q^' ^ with a radius equal to the mean radius of the cutter minus one- 
half the thickness of the cutter blade at its apex from the points I" and m" — 
the latter points being located as were the points V and m' , 

Finally, with O as a center, the points (^ and Q'' are swung over to the 
plane of vertical settings and the respective vertical settings, X' and X'\ 
carefully scaled. The vertical settings thus secured differ from those 
obtained by the formula method as the three points Q, Q' and Q" do not lie on 
the arc of a circle having point O as a center. In some cases — for instance, 
on the layout depicted on Fig. 253 — the vertical setting for the convex side 
of the tooth may be greater than that for the concave side when the settings 
are obtained by the layout method, which would never occur in the formula 
method. 

COMPARISON OF METHODS 

The relative accuracy of the two methods is open to argument. The two 
sides of the teeth being finished with slightly different radii of cutter, there 




Horiiorrf-al 
Axis of ±. 
Gear Blank 

Po/'nis ofTangena/^- 
Tooth Profiles and 
•Spiral Anak Plane 



FIG. 254. 



can actually be but one point of perfect tooth contact, but for all practical 
purposes contact takes place over quite an appreciable length of tooth, even 
before the gears have been run together and worn into perfect mesh, as the 



3i6 AMERICAN MACHINIST GEAR BOOK 

difference in the heights of the respective arcs — that of the concave side and 
that of the convex side of meshing teeth — is extremely shght. In gears 
finished on vertical settings arrived at by the formula method, the point of 
perfect contact is located on both convex and concave profiles at the same 
relative point — on the mid-pitch circumference where the curved profiles of 
the teeth are tangent to the spiral angle plane. In gears finished on vertical 
settings arrived at by the layout method the point of tangency between the 
spiral angle plane and the tooth profile is not located at the same relative 
point on both the concave and convex profiles, the points of tangency 
depending upon the relation of the horizontal setting to the vertical settings, 
the angle of spiral, etc. (see Fig. 254). 

When the more remote point of tangency is on the convex profile, as 
depicted on Fig. 254, this failure of the spiral angle plane to be tangent to 
both sides of the tooth at the same relative point is actually a slight advant- 
age as it locates the point of contact at a point where the power transmitted is 
slightly greater than at mid-tooth and where the tooth is slightly heavier. 
In cases where the contact point on the convex profile is slightly nearer the 
inner edge of the gear than mid-tooth no appreciable disadvantage exists on 
account of the flatness of the contact arcs. The slight eccentricity given to 
the load on the teeth by the failure of the spiral angle plane to be tangent at 
corresponding points on both sides of the tooth tends to accelerate the slight 
weave of parts which produces the perfect tooth contact of spiral type bevel 
gears which have been run together. 

The layout method is open to the possibility of error in scaling and laying 
out the diagram and, in addition, consumes considerably more time than the 
simple calculations required for the formula method. This excess time could 
be put to far better advantage in running the gears together to perfect tooth 
contact — at least, the time saved by the formula method would compensate 
for the longer time required in truing up the tooth contact of gears generated 
on vertical settings derived from the formulas. Both methods of arriving 
at the vertical settings for the finishing cuts have their advocates so that the 
choice of methods is almost entirely a matter of personal preference on the 
part of the manufacturer or machinist. 

WILLIAMS "MASTER FORM*' BEVEL GEARING 

In Section V mention was made of the radical departure from the so-called 
standard proportions of bevel gear teeth evolved in Pentz ''parallel depth" 
bevel gears, in which the gear teeth are of constant depth. This innovation 
was brought about chiefly to reduce the cost of cutting bevel gears, the 
efficient bevel gear generating machine of today not having been perfected 
at that time. Eliminating the necessity of tapering the depth of the teeth 
unquestionably simplified gear cutting, but the complications incident to the 
variation in profile curvature of the teeth, from end to end of tooth, remained 



SPECIAL BEVEL GEARS 



317 



and necessitated the use of formed cutters of special cutting profile to finish 
the gears and a multipUcity of adjustments of the gear blank on the work 
arbor. 

In the Williams "Master Form" System of Bevel Gearing, developed by 
Harvey D. Williams, the radical step is taken of avoiding the variations in 
tooth profile in the larger of a pair of gears — the gear, rather than the pinion 
— by making the tooth spaces simply straight gashes across the face of the 
gear. The "master form" feature of such gears is that the tooth profiles are 
straight, Hke those of the famiHar involute rack teeth. The thickness of the 
teeth varies from end to end, owing to the difference in the inner and outer 
circumferences of the gears, but the tooth spaces are uniform as to depth and 
section. 

The pinion meshing with this "master form" gear is a generated member. 
That is, its teeth have curved profiles which are most accurately and cheaply 

K a {////// //a \ 

Detail of GearTee+h 





FIG. 255. WILLIAMS MASTER FORM BEVEL GEARING. 

produced by the generating principle with cutting tools which are the con- 
jugate of the simple rack tooth employed in gashing out the tooth spaces in 
the "master form" gear. 

This radical departure from the usual design of bevel gearing (see Fig. 
255) is particularly noteworthy in that it is a reversion to and adaptation of 
fundamental mechanical principles which appear to have been overlooked 
until quite recently in the evolution of gearing — spur gearing as well as bevel 
gearing. Toothed gears will roll together smoothly and efficiently provided 
their teeth and tooth spaces are conjugately related. This is the one essen- 
tial requirement and the form and shape of the teeth is of quite secondary 
importance. 

The development of the almost universal octoid system for bevel gearing, 
with its varying tooth profile, not only from end to end of tooth, but for 
every size of gear, is based on the requirement that each and every bevel 
gear be conjugately related to a crown gear with flat sided teeth. This is in 
line with the simple rack tooth base for the involute system of spur gearing, 
and though there may be some advantage to such a basis in spur gearing — in 



3i8 AMERICAN MACHINIST GEAR BOOK 

commercial gear production in large quantities this is now recognized as 
more or less of a fallacy — there is none in the case of bevel gearing. 

In the Williams '* Master Form " bevel gear, the teeth are almost as readily 
cut as those of an ordinary spur rack and more expeditiously and cheaply 
than they can be generated. Each gear cut with these "master form" 
teeth becomes the standard to which its mating pinion or pinions are pro- 
portioned. As these pinions can be generated as readily as those with octoid 
teeth no added difficulty of expense is entailed in the manufacture of the 
pinions. 

PRODUCTION ECONOMIES 

The cost of cutting Williams ''Master Form " bevel gears depends largely 
upon the number of teeth on the gears and it has been estimated that about 
80 per cent, of the combined number of teeth on the gear and pinions 
of automobile transmissions — for which mechanisms these distinctive gears 
are chiefly used — are on the gear, or larger wheel. The Williams system 
materially reduces the cost of cutting the teeth on the gear, resulting in a 
considerable reduction in the cost of manufacturing the complete transmis- 
sion, as the cost of generating the Williams pinion is no greater than that of 
generating an octoid pinion of the same size. 

The economy in manufacturing cost is probably the chief advantage 
of the Williams system of bevel gearing, but it also possesses other features 
which have considerable merit. The peculiar shape of the teeth makes the 
gears exceedingly smooth running and they are said to develop unusually 
high efficiency in operation. Another advantage of the gears — one which is 
of particular value in automobile drives — is the distinctive capacity of the 
gears to discount disalignment. 

EFFECT OF SPRUNG SHAFTS 

The strains and wrenches to which the transmission of an automobile is 
frequently unavoidably subjected have a tendency to spring the driving shaft 
out of alignment, seriously interfering with the satisfactory operation of the 
transmission, which depends largely upon the exactitude of shaft alignment. 
The teeth of the gear may remain in mesh with those of the pinion, but the 
sprung shaft throws a heavy unbalanced pressure on the gears, causing noisy 
operation and undue tooth wear. This is well nigh unavoidable whatever 
system of gearing is employed, but with Williams gearing the troubles 
produced by sprung shafts are greatly discounted. 

Fig. 256 shows diagrammatically what might be expected to occur should 
the transmission shaft of an automobile be sprung out of alignment, from 
O to 0\ The normal position of the gears with their centers at O is depicted 
in full lines and in sprung position with centers at O'. For the gears to 
remain in mesh, the inner corner of the tooth, A , remains in the same relative 
position, but if it were not for the restraining influence of the meshing teeth 



SPECIAL BEVEL GEARS 



319 



the diagonally opposite corner would move from B to B' and the strain to 
which the gearing is subjected in preventing such movement is measured by 
the distance between B and B' . In the case of the ordinary octoid form of 




Co 



o 





!^^ 




\ 

A 


ir\ 


^ 


~-^ 


' 


1 


\ 
\ 


1 

/ 
/ 

; 
/ 


/ 1 

/ 
/ 
/ 




1 

1 

1 
1 


-^ 


1 
\ 




! 
I 
1 


'-^^l 


"\ 





FIG. 256. EFFECTS OF SHAFT DISALIGNMENT. 

tooth, the disalignment strain is measured by the distance X and in the case 
of the Williams ''Master Form" of tooth by the distance Y, and Y is very 
appreciably less than X. The difference in intensity of strain, furthermore, 
is much more than the difference between the points B and B' for the respec- 
tive strains and, if the gears are under considerable load, may produce fracture 
in one case and not in the other. 



SECTION XV 

Williams System of Internal Gearing* 

The advantages of internal gearing may be summarized briefly as a 
compactness of design impossible of realization with externally meshing 
gears, an important safety feature in that the gear forms its own gear guard, 
tooth contact over a considerably longer arc than in either external or rack 
gearing, a reduced sliding action between meshing teeth, improved operating 
action, reduced wear and a considerably longer life of usefulness than can be 
possessed by externally engaging gears. 

With such a formidable list of advantages favoring the internal gear, its 
chief limitations in the past have been the difflculty and cost of accurately 
cutting the internal gear teeth and the fact that true involute profile is rarely 
practical or feasible for the internal gear teeth, on account of the interference 
developed with the teeth of mating pinions. The difficulty and cost of 
cutting the internal gear teeth have been overcome, to a considerable extent, 
by the development of generating machines of approved type for cutting 
gear teeth, but the limitation set by interference — "fouling" of teeth — has 
only been reduced by marked modifications of the involute profile of the gear 
teeth. That is. the profiles of "standard" involute tooth — the form of 
tooth almost exclusively employed for internal gears — diverge to a marked 
degree from the true involute curve for the greater part of their depth. 
The necessary modification of the tooth profile causes the "standard" 
involute tooth for internal gears to lose to a large extent one advantage the 
involute system of gearing is supposed to possess — that of interchangeability. 

LIMITATIONS OF INVOLUTE INTERNAL GEARING 

A more serious limitation placed upon internal gearing by the involute 
system is that, despite the modification to the profiles of the gear teeth, there 
is a practical limit to the minimum number of teeth an involute gear may 
have — generally taken as 12 — so that the smallest feasible pinion must have 
that number of teeth. For a 12-tooth pinion with teeth of involute form to 
possess the required strength, it is quite frequently necessary to employ a 
heavy pressure angle and adopt the stub form of tooth. 

Increasing the pressure angle tends to increase also the arc of contact, by 
enlarging the angle of approach, but this gain is secured only through a 
certain sacrifice in the efficiency of power transmission. The heavier pressure 
angle increases the axial, or radial, component of the power transmitted and 
decreases the effective component. 

The involute system of internal gearing and its limitations have been 

touched upon before discussing the Williams system of internal gearing to 

* From a report on the system by Reginald Trautschold. 

320 



INTERNAL GEARING 



321 



show that profiles of the internal gear teeth cannot be produced in commercial 
gearing of true involute form and the questionable advantage of interchangea- 
bility has to be sacrificed in any case, the limitation in the minimum number 
of teeth and the loss in transmission efficiency entailed in the use of the heavy 
pressure angle necessary for an involute pinion to possess adequate strength. 



- Paih o-f PoiniofCon^ac^ 




h = 



90 



FIG. 257. WILLIAMS SYSTEM OF INTERNAL GEARING. 
FORMULAS 

a' = c -\- b L = o.sD' sin. a' 



R = 0.2^ D' COS. a' A = R sin. a' B =o.^D' — Rcos. a' 

^ T ^ Z7r>- J? ^ - ^ 

tan. w = -j^ I an. X = -p^ ^ h = 2R sm. w t = 

B C — B tan. w 

E 



tan. y = 



F -G 



z = X -\- y 



322 AMERICAN MACHINIST GEAR BOOK 

WILLIAMS SYSTEM OF INTERNAL GEARING 

In the Williams system of internal gearing, the forms of the gear teeth in 
both the internal gear and the mating pinion depart radically from those 
designed according to other systems but, in so doing secure certain inherent 
advantages that give to the gearing an improved operating action, increase 
the strength of the pinion teeth, reduce wear and greatly reduce the cost of 
producing accurate and efficient internal gearing of the spur or spiral type. 
The limitations placed upon internal gearing designed according to other 
systems are materially reduced, while all the advantages are retained and 
accentuated. 

THE WILLIAMS SYSTEM 

In the Williams system, the teeth of the internal gear are made with 
straight line profiles — the bounding profiles of the tooth spaces being similar to 
those of an involute rack — while the teeth of the mating pinion are provided 
with curved profiles of conjugate form, giving a combination that results in 
a distinctive path to the point of contact between the gear and pinion teeth 
throughout their arc of contact (see Fig. 257). An extension of each flat 
profile plane — the profile planes of the internal gear teeth — is tangent to a 
circle concentric with the circumference of the internal gear and the radial 
normal to a tangent from any point on the path of point of contact will, 
with the radial line to such point of contact, form a right angle triangle. The 
path traced by the apex of such triangle is over that section of the path of 
point of contact that can be made of use in practical gear combinations and is 
— within the limits set by attainable accuracy in commercial manufacture — 
the arc of a circle of a radius equal to one-half the length of the tangent from 
the instant axis, m, of the gears to the concentric circle of tangency, about its 
mid-point. That is, the usable section of the path of point of contact is, to 
all intents and purposes, the arc of a circle passing normally through the 
instant axis of the gears and tangent to the radial normal to the profile plane 
passing through the instant axis. The point on this path of point of contact 
at which contact can first occur is located by the radius of the circle of path 
of point of contact passing through the center of the pinion mating with the 
internal gear, and the point at which contact must cease is fixed by a radial 
line from the center of the internal gear passing through the center of the 
circle of path of point of contact. Referring to Fig. 257, in which the rotation 
of the gears is in a clockwise direction, contact between pinion and gear 
teeth — if the teeth were suitably proportioned — would commence at point 
q, on the path of point of contact, and end at the point r. The angle of 
approach, in such a case, would be measured by the angle x and the angle of 
recession by the angle y, or the total arc of pinion contact measured by the 
angle e. 

In the combination illustrated, contact does not commence as soon, nor 



INTERNAL GEARING 



323 



is it maintained so long. Contact begins at the point at which the inner 
circle of the internal gear crosses the usable path of point of contact and ends 
at the point at which the outer circle of the pinion crosses the same path. 
Despite the failure to utilize the maximum usable section of the path of point 
of contact, the actual arc of pinion contact is unusually long for the gearing 
combination shown. The arc of contact is, in fact, materially longer than 
that securable with gear and pinion teeth proportioned according to any other 
of the systems of internal gearing in common usage. To appreciate this 
important fact in full, a comparison of the paths of point of contact for inter- 
nal gearing combinations of similar ratios, but designed according to other 
systems of tooth form, is necessary. 

PATHS OF POINT OF CONTACT 

The four diagrams shown in Fig. 258 illustrate in a graphic and illuminat- 
ing manner the paths of point of contact in the principle standardized systems 
of internal gearing. In each instance, the direction of rotation in clockwise 



Mh of Poinhf Confacf 
20 Otgne Pressure 
_.■■-■ An^h 

" ^ -^ -'PM o fPo/nf fff: ; 

Corrhcf-I^j Decree Pressure 



..Paih of Pami-cf Conhd- ?0 Degrees 
Obliguifu. ^ 




Mh of Piiniof Con hcf-- 
14 1 Degreti 
J^ ObliquittJ. 



...■PaHiofPoiniof 
/ Conincf 





-4- ' ^ -" ^ ' 4< 

FIG. 258. COMPARISON OF PATHS OF POINT OF CONTACT. 

and the shaded area between the pitch circles of the gears and pinions — A 
and B respectively — indicate areas within which no part of the path of point 
of contact can lie, for in such areas interference between gear and pinion 
teeth would result. 

Diagram (I) depicts a combination of gears, in which the point of contact 
is concentrated and localized at the instant axis, Z, or on the pitch circles of 
the gears. If it were possible to devise a system of toothed gearing in which 
the path of point of contact was so localized on the pitch circles, it would 
result in concentration of wear at a point first on the pinion tooth, say on the 
section of the pitch circle from a to Z and then on the internal gear tooth, on 
its pitch circle from Z to b but, at the same time, a path of point of contact of 
a long arc of action. The long arc of action would be a decided advantage, 



324 AMERICAN MACHINIST GEAR BOOK 

but the concentration of wear a decided disadvantage. A radial path of 
point of contact passing through the instant axis, as every path of point of 
contact must do, would give no arc of action at all. However, these hypo- 
thetical paths of point of contact, the one following first the pitch circle of 
the pinion and then that of the internal gear and the radial path passing 
through the instant axis, represent the maximum and minimum arcs of tooth 
contact. 

The path of point of contact shown in diagram (II) illustrates that of the 
epicycloidal system of gear teeth. The profiles of the teeth formed in this 
system are developed by a point on a generating circle rolling first on one 
side and then on the other of the pitch circles, so that path of point of contact 
is a curve normal to such profiles and similar to that depicted as aZh. The 
epicycloidal system of forming gear teeth has now been virtually abandoned 
except for special combinations of internal gearing where the gear and pinion 
have very nearly the same number of teeth and the form of involute teeth 
would make it impossible for such teeth to clear themselves, on account of the 
cost and difficulty of accurately proportioning the teeth. Furthermore, the 
diagram shows that about the instant axis of the gears the epicycloidal path 
of point of contact closely follows the pitch circles, concentrating the wear 
on the teeth — particularly in the case of the pinion. At the same time, the 
extremities of the path of point of contact tend to approach a radial direction 
and so curtail the length of the arc of contact, or action. A limited arc of 
contact and concentration of wear at and near the pitch circles, as well as 
the cost and difficulty of accurately forming the teeth, well justify the general 
abandonment of the epicycloidal system of internal gearing. 

In diagram (III) are depicted the paths of point of contact for 143^^-degree 
and 20-degree teeth in the involute system of gears, aZh and a'Zh' respec- 
tively. The paths are flat planes passing through the in stant axis and inclined 
to the common tangent plane at the respective pressure angles. The com- 
mencement of tooth contact on the flank of the pinion, and therefore the 
limiting factor in the useful depth of the gear teeth face, is fixed by the pinion 
radius normal to the approach section of the path of point of contact, while 
the limit to the recession section of the path of point of contact is — theoret- 
ically, at least — unlimited. However, there is a practical limit set to the 
length of the recession section of the path of point of contact by the length 
of the feasible pinion tooth addendum, for if the flank of the internal gear 
teeth is too long the tops of the pinion teeth are cut away by the conjugate 
profiles of the mating gear. 

The quite general adoption of 20-degree involute teeth for internal gears 
has been brought about by an attempt to secure a longer arc of approach than 
can be secured with i4j^-degree involute teeth. In spite of such practice, 
the angle of approach remains quite limited and an effective angle of contact 
for involute gears is to be secured only through adopting the stub type of 
tooth with heavy pressure angle for the internal gear and decreasing the 



INTERNAL GEARING 325 

addendum of the pinion teeth, upon the face of which the wear is more 
concentrated than it is upon the flank of the internal gear teeth. 

The fourth of the diagrams illustrates the paths of point of contact for 
14^-degree and 20-degree obliquity teeth at the instant axis formed according 
to the Wilhams system, aZb and a'Zb' respectively. The much greater 
approach section to the path of point of contact is very apparent and well 
typifies the reason for the improved operating action of the WilHams gear 
A Wilhams gear tooth of ^K-degree obliquity has a considerably longer arc 
of approach than has a 20-degree involute tooth and the usable section of the 
recession path of point of contact for the i4K-degree Williams tooth is also 
much longer than can be made of use for a 20-degree involute tooth In the 
case of 20-degree Wilhams tooth, the increase in length of arc of action is as 
pronounced as m the case of involute teeth. 

It will also be noted that the wear on the teeth is quite evidently much 
more uniformly distributed in the Williams system of gearing than it is in 
the involute and, furthermore, that such concentration of wear as does occur 
IS toward the bottom of the flanks of both pinion and gear teeth, where the 
greatest amount of metal is and where, owing to the distinctive form of the 
teeth, wear can occur with less sacrifice of tooth strength than in any other 
form of tooth. Wear at such points will also have the tendencv to ease off 
the shock of imtial contact and. so produce quieter and smoother operating 
gears. ^ 

STRENGTH OF PINION TEETH 

In the matter of strength, the pinions-always the weaker of a pair of 
mating gears-proportioned according to the Williams system are consider- 
ably superior to those of equivalent involute form and this pecuKarity is 
clearly demonstrated by a comparison of the forms of the respective internal 
gear teeth. The teeth of the Williams pinion have curved profiles of exact 
conjugate form for the particular Wilhams internal gear with which it is to 
operate, while the accurate curves of the profiles of the involute pinion teeth 
are generated by the same or similar involute rack tool employed for the 
generation of the mating internal gear. 

The tooth forms illustrated in Fig. 259 are accurate representations of- 
(a) involute mternal gear teeth; (b) an accuratelv mating pinion tooth- (c) 
equivalent Wilhams internal gear teeth; (d) a mating Williams pinion tooth- 
and (e) the two forms of pinion teeth superimposed to emphasize the differ- 
ence m their respective tooth profiles. The curved profiles of the involute 
internal gear teeth, both above and below the pitch circle, it will be noted are 
concave and so curve away from the contact planes tangent to the tooth 
profiles on the pitch line. The straight profiles of the Williams internal gear 
teeth, on the other hand, lie in the contact planes of the gear teeth The 
extra root thickness of the involute internal gear tooth, though adding to its 
strength, results m a corresponding decrease in the thickness of the top of 



;26 



AMERICAN MACHINIST GEAR BOOK 



the mating pinion tooth, which though perhaps not weakening the tooth does 
tend to Umit the depth of its addendum. The extra top thickness of the 
gear tooth, however, has a more serious effect, for it tends to weaken the pin- 
ion tooth by the necessary undercutting required for clearance. The com- 
posite diagram (e) forcibly illustrates the strength superiority of the 
Williams pinion tooth, for though the pinion teeth selected for comparison 
have been chosen as only slightly undercut — the involute tooth is a 20-degree 
80 per cent, stub tooth of a 12-tooth pinion — the difference in their root 





--. Wlliams Pinion Toofh 
(Full Line Oirhhne) 



Involu-h Pinion Tool-h 
"^ {DofhdLinz OuHme) 
I 



20 Dz^rees Pressure 

Angle 

Speed Rah 0-20=12 



FIG. 259. COMPARISON OF TOOTH FORMS. 



thicknesses is quite sufficient to indicate an appreciable difference in the 
strength of the two types of pinion teeth. The strength of the respective 
teeth are proportional to the square of their root thicknesses. 

The proportions of Williams internal gear teeth, furthermore, are such as 
to permit a modification in customary design whereby the strength ol the 
gearing — measured by the strength of the pinion teeth — is materially 
increased, by making the pinion teeth as strong as the gear teeth. This is 
accompHshed, as illustrated by the eight tooth (6 to i) combination shown 
in Fig. 260, by increasing the thickness of the pinion teeth and correspond- 
ingly decreasing the thickness of the gear teeth, so the root thicknesses of the 
respective teeth are equal. The pitch of the gears is not altered, simply the 



INTERNAL GEARING 327 

thickness of the pinion teeth increased and the space between adjacent gear 
teeth widened by reducing the thickness of the gear teeth. 

This method of securing a high-speed ratio with a pinion of few and 
unusually strong teeth cannot be resorted to as effectively in the case of gears 
proportioned according to the involute system. In gearing so proportioned, 
the undercutting of the pinion with few teeth, though they be amply thick 
on the pitch circle, would be so excessive as to necessitate unduly reducing 
the thickness of the gear teeth to equalize the strength of the pinion and gear 
teeth. 

WEAR 

The superiority of the Williams system of internal gearing in the matter 
of longer arc of contact, or action, is well demonstrated by a graphic com- 
parison of similar gear combinations of Williams and involute form. Fig. 
261 diagrammatically illustrates the operating relation between the Williams 
internal gear and its mating pinion shown in Fig. 257, and also the operating 
relations for involute gears of the same ratio and proportions. 

The section of the path of point of contact actually utilized in this com- 
bination of Williams gears is from the point 5 to the point /, or the section 
included between the intersections of the path of point of contact and the 
inner circle of the internal gear and the outer circle of the mating pinion 
respectively. The approach of the pinion teeth is measured by the angle e 
and their recession by the angle/, or the length of the pinion arc of contact, 
or action, is measured by the angle g. These angles, it will be noted, are 
materially less than the usable angles of approach and recession as limited by 
the available arc of contact, fixed by the extreme points q and r of the path 
of point of contact, yet they are considerably greater than the equivalent 
angles for an involute gear combination of the same speed ratio and pressure 
angle — i,e.^ when the addendum and dedendum dimensions of the teeth are 
the same. The oblique line tangent to the Williams path of point of contact 
at the instant axis of the gears, the line s't' ^ depicts the path of point of con- 
tact for such involute gear combination. Contact between opposing involute 
profiles cannot commence in advance of the point 5', the intersection of the 
path of point contact with its radial normal from the pinion center, and can- 
not continue beyond the point t\ where the outer circle of the pinion crosses 
the path of point of contact. 

In the involute gear combination, the angle of pinion approach, a, is 
equal to the pressure angle and is quite noticeably less than the angle of 
approach, e, for the Williams gear combination. The angle of recession, k, 
is also less than the corresponding angle for the WilHams gear combination, 
so that the total pinion angle of contact, /, or action, is, in the involute gear 
combination, only about 85 per cent, of its value in the case of the Williams 
gears. Furthermore, the delayed commencement of contact in the involute 



328 



AMERICAN MACHINIST GEAR BOOK 



combination limits the tooth profile area over which contact occurs to 
approximately 75 per cent, of that so utilized in the more efficient construction. 
With an arc of contact, or action, 173^ per cent, greater and 50 per cent, 
more tooth profile area over which contact between the mating teeth takes 
place, the Williams gear combination has a wear resisting capacity some 75 

Paih of Poinhf Conhc^ 



'\. 48 Toofh Inh\naf diar 
r 




\ 
. \ 



\ 
\\ 



\\ 



\ 



— ,^- 

FIG. 260. 6 TO I WILLIAMS INTERNAL GEAR COMBINATION. 

per cent, greater than the similar involute gear combination, when trans- 
mitting the same power at the same speed of rotation — provided, of course, 
that the respective gears are constructed of materials of similar wear resist- 
ance. These comparative values are, naturally, applicable only to the 
particular gears under consideration and will vary to some extent for other 
gear combinations, but they are practical. 



INTERNAL GEARING 



329 



To secure a utilizaton of the full depth of the involute tooth profiles for 
contact, in a combination such as that depicted in Figs, i and 5, would 
necessitate increasing the pressure angle to something over 20 degrees — the 
obliquity of pressure at the instant axis for the gears as illustrated being 143^^ 
degrees — and though such modification would "ncrease the angle of pinion 
approach to about that for the i4j^-degree Williams gears, the angle of 



H M >y N ■ ,>\ 










FIG. 261. COMPARISON OF ARCS OF CONTACT. 



recession would be quite materially reduced, unless the flank of the gear 
teeth was considerably increased. Such modifications would be too excessive 
to be feasible, but if they could be made and were also adopted for the 
WilUams gear combination, the superiority of the latter form of construction 
in the matter of angle of contact would still be marked, and if the flank of 
the pinion teeth w^as also increased — which could be profitably done in the 
Williams system, but not in the involute — the superiority of the Williams 
construction in lengthened arc of contact and also increased wear resistance 
would be substantially the same as in the gear combinations that have been 
discussed. 



330 



AMERICAN MACHINIST GEAR BOOK 
REDUCTION IN THE NUMBER OF PINION TEETH 



The teeth of the pinion in a Williams combination are proportioned to 
operate with a particular gear, or, in other words, the pitch and obliquity of 
the internal gear teeth positively control the profile curvature of the mating 
pinion teeth. It is obvious, therefore, that pinions with fewer teeth than are 



Mho-FPoi'niof 
Corrf-ad--., 

777777^7777. 




FIG. 262. 5 TO I WILLIAMS INTERNAL GEAR COMBINATION. 

feasible in the involute system of gearing can be employed, for if they can be 
generated at all, they will operate with their "master" gear without inter- 
ference of teeth. Gear combinations with pinions with a small number of 
teeth are both feasible and practical in the Williams system of internal 
gearing. 

Fig. 262 illustrates a combination of Williams gears — a 5 to i ratio — 
in which the pinion has only six teeth, yet they are well proportioned for 
strength and free from objectionable undercutting. The obliquity of the 
gear teeth is marked, it is true, but the reduction in the efficiency of power 
transmisson is far less than it would be in the case of involute gears with a 
correspondingly heavy pressure angle, for during the recession period of the 



INTERNAL GEARING 331 

engaging teeth — the portion of tooth contact during which the transmission 
of power from the pinion to the gear is considerably more effective than 
during the approach to the position of so-called full mesh — the obliquity of 
the direction of pressure reduces steadily. 

The gear combination shown in Fig. 260 illustrates a method of pro- 
portioning the pinion and gear teeth by which a pinion with only eight teeth 
was employed in a 6 to i reduction without sacrifice of strength or adopting a 
tooth of unusual obHquity. 

This distinctive feature of ability to generate and use pinions with a 
small number of teeth is quite obviously an important advantage of the 
Williams system of internal gearing. Greater speed ratios are obtainable 
without unduly increasing the diameter of the internal gear, or a coarser 
pitch may be employed than is feasible with involute gears, when available 
space for the accommodation of the gearing is limited. 

The ability to employ gears with fewer teeth of heavier pitch than can be 
used in the involute system — practice, always to be recommended — is 
particularly advantageous in the Williams system on account of its 
superiority in length of arc of tooth contact and improved operating action. 

MACHINING TEETH 

The present increasing demand for internal gears of the involute form has 
been brought about largely by the perfection of efficient and accurate gear 
cutting or generating machines, machines which automatically modify the 
involute profile of the gear teeth to decrease in large measure the limitations 
set by the interference of teeth of true involute profile. In the Williams 
system of internal gearing, modifications of tooth profiles are unnecessary, 
for the pinion is developed to operate with a particular mating internal gear 
and only with such or similar gear. In this manner, interference in gear 
combinations for which commercial demand exists is virtually eliminated and 
no provisions have to be made in machining the gears. 

The straight tooth profiles of the Williams internal gear make the accurate 
cutting of such gear teeth a comparatively simple and inexpensive operation. 
A plain "V" shaped, reciprocating cutting tool will gash out the tooth spaces 
rapidly and finish accurately both of the bounding tooth profiles, necessitat- 
ing only the simple adjustment act of indexing the gear blank from tooth 
space to tooth space. Any ordinary shaper with a suitable indexing mechan- 
ism can be employed. If the internal gear is of the ring type, the tooth 
spaces can be milled with even greater rapidity, the cutter — as in the case 
of the simple reciprocating tool — ^being easily ground with the utmost 
precision as to the accuracy of the cutting edge and its obliquity. 

The machining of the pinion teeth is somewhat more complicated, as 
they either have to be generated or a cutting tool employed, conforming 
in profile to the outline of the tooth space by being accurately fitted to a 
generated pinion tooth space template. The generation of the pinion teeth 



332 AMERICAN MACHINIST GEAR BOOK 

or of the pinion teeth space template requires the use of a simple generating 
machine in which the simple cutting tool, conforming in shape to the straight 
profile internal gear tooth, swings about the rotating pinion blank on a radius 
equal to one-half the pitch diameter of the mating internal gear, so that the 
ratio of the rotating speeds of the cutter and pinion blank is the same as 
that of the speed ratio of the gear combination in which the pinion is to be 
employed. As the pinion of the ordinary internal gear combination has 
comparatively few teeth their generation is a, simple task compared to the 
task of generating the teeth of the mating gear, but when a large number of 
pinions are required of a specific size to operate with internal gears of given 
size and proportions, as in the manufacture of gears for a standardized 
product produced in quantities, a formed cutter, fitted to a generated pinion 
tooth space template, will be found to expedite materially the cutting of the 
pinion teeth. 

COSTS OF MACHINING WILLIAMS GEARS 

The cost of cutting Williams internal gears and gear combinations is 
quite obviously very much less than that for machining gears of other tooth 
form, and is more and more marked as the difference in the number of teeth 
in the internal gears and their mating pinions increases. If the average 
speed ratio of internal gear combinations is taken as 4 to i , the relative time 
consumed in cutting a gear combination proportioned according to the 
Williams system, compared to that required to generate similar gears pro- 
portioned according to the involute system, is substantially as follows: 
When generating all the Williams pinion teeth, 50 to 60 per cent., and when 
cutting the Williams pinion teeth with a formed cutter, 30 to 40 per cent. 
In addition to this very material saving in the time required to machine the 
gear teeth, less skilled operators may be employed for the simple operations 
entailed in cutting Williams gears than can be safely trusted to operate the 
more intricate machines required for generating the involute type of tooth, 
so that the total cost of manufacturing Williams internal gears and mating 
pinions is very substantially less than that of generating similar gears 
designed according to the involute system of gearing. 

DISTINCTIVE ADVANTAGES 

The advantages of the internal type of gearing are now quite generally 
appreciated and conceded, and in every instance the superiority of the Wil- 
liams system of internal gearing is conspicuous. Briefly summarized, the 
more evident of the advantages that are possessed to an accentuated degree 
by the gear combinations proportioned according to this system are: 

1. Greater length of arc of tooth contact, due to the distinctive path of 
point of contact. 

2. Reduction in wear, due to the prolonged contact and the greater 
proportion of the tooth profile areas utilized for contact. 



INTERNAL GEARING 7,7,7, 

3. Increased strength of pinion teeth, on account of the reduction in 
undercutting. 

4. Improved operating action, due to the sHght relief in shock of tooth 
contact occasioned by the concentration of wear on the flanks of the pinion 
and internal gear teeth. 

5. Reduction in the required number of pinion teeth. 

6. Reduction in the diameter of the internal gear for a given speed ratio 
and load. 

7. Greater speed ratios without increase in diameter of internal gear. 

8. Possibility of employing a coarser pitch without increasing the diam- 
eter of the internal gear. 

9. Increased strength of gear combinations. 

10. Simplicity of design. 

11. Accuracy and ease of gear tooth reproduction. 

12. Greatly reduced cost of manufacture. 



SECTION XVI 

Rolled Gearing* 

The evolution in commercial gear production which entails the simple 
process of forging teeth on the gear blanks heated to a semiplastic condition, 
by causing them to roll under heavy pressure in positive synchronized con- 
tact with hardened die rolls, is unquestionably the outstanding achievement 
in modern methods, for not only is a much lower cost of production realized, 
but the gears produced are stronger, tougher and more accurately formed 
than the best types of gears produced by machining processes of generation. 
The process involved is essentially one of molding generation and the gears so 
forged operate with high degree of pure rolling action, insuring their high 
mechanical efhciency in service. To appreciate the validity of these claims 
necessitates a clear understanding of approved gear cutting methods. 

Originally the machining of toothed gearing involved the operation of 
cutting each individual tooth space with the aid of a formed cutter, so that 
the section of the tooth space was conjugately related to that of the teeth 
formed on the gear blanks. The next step in the development of modern 
gear production practice was the evolution of the so-called generation of gear 
teeth, whereby an attempt was made to duplicate in the production process 
the action of engaging gears, by causing the cutter and the work to advance 
as if in mesh while the tooth spaces were machined. The gear shaper, in 
which the cutter takes the form of a pinion which is made to cross the face 
of the gear blank rapidly — removing metal on the forward stroke — while the 
cutter and blank roll together much as if they were pinion and gear, and the 
gear hobber, in which the teeth of the hob are patterned after the teeth of an 
involute rack and the gear blank and hob are caused to move together much 
as a mounted gear would be actuated by an advancing rack with which it 
was in mesh, illustrate the approach made toward a simplification of process 
by attempting to employ the same simple rolling action, which is the objective 
of gearing, in the process of gear fabrication. 

WEAKNESSES OF GEAR GENERATION 

Although these methods of gear generation exemplify very decided 
advancement in the art of producing high grade gearing, both economically 
and expeditiously, they possess certain weaknesses which in the light of more 
recent developments limit the precision with which the gear teeth may be 
formed and also entail considerable scrapping of metal. These weaknesses 
"^ Reginald Trautschold, Consulting Engineer, The Anderson Rolled Gear Co. 

334 



ROLLED GEARING 335 

are inherent respectively to the generation method of machining and to 
machining operations in general. 

The generation method entails two quite unrelated operations, the revolu- 
tion of the gear blank as if actuated by a pinion conforming in tooth dimen- 
sions to the dimensions of the cutting tool and the travel of the cutting tool 
across the face of the gear blank. These two operations cannot be performed 
simultaneously without some sacrifice in the accuracy of tooth formation. 
Since neither the shaping nor the hobbing process can give a continuous 
amount of metal, else there could be no advancement of the gear blank, a 
series of more or less pronounced ridges and flats mar the surfaces of ''gen- 
erated" gear teeth. 

Another weakness common to all machined, or cut, gearing is the unavoid- 
able removal of metal to form the tooth spaces of the gears. The metal 
so removed, though of high grade, has no value except a quite nominal one 
as scrap. 

DEVELOPMENT IN GEAR ROLLING 

The method of overcoming these drawbacks, or weaknesses, to the genera- 
tion method of gear cutting, is by rolling the teeth on gear blanks heated to 
an effective forging temperature by means of a hardened die roll. This 
process is not a new discovery, but its practical development for commercial 
gear production is distinctly so. As early as 1872, John Comley attempted 
to roll teeth on hot gear blanks by a simple knurling process and since that 
date there have been a number of similar attempts, but none of these experi- 
menters succeeded. In each case, knurling was employed — that is, either 
the shaft carrying the forming die or that supporting the heated gear blank 
was positively driven, but not both — with the result that though teeth were 
produced they were malformed and poorly spaced. The failures were due 
to unavoidable slippage between the heated gear blanks and the die rolls. 
Nevertheless the attempts were highly valuable in the light of later develop- 
ment as they demonstrated the entire feasibility of the rolling process. 
They proved that by rendering gear blanks sufficiently plastic by heat to 
permit the forging of teeth on their peripheries, a simple process of rolling 
the heated gear blank with a hardened steel die roll constituted a practical 
means of gear tooth formation. 

It remained for H. N. Anderson, however, to conceive and develop a 
practical method of accomplishing this result and he did so by the simple 
expedient of driving positively, through substantial timing gears, both the 
shaft carrying the heated gear blank and that supporting the die roll. In 
this manner the slippage between die roll and blank which caused the failures 
of the earlier attempts to roll gears by a simple knurling process was entirely 
eliminated and the teeth rolled on the blank accurately spaced and well 
shaped. 

The first gear rolling machine to be constructed on this principle was 



336 , AMERICAN MACHINIST GEAR BOOK 

completed in 191 1 for the intended purpose of employing the gear tooth roll- 
ing method simply to supplant the tooth gashing operation in cut gear pro- 
duction then in general use. However, the teeth rolled by this pioneer 
machine were found to be so well and accurately proportioned and formed 
that any subsequent machining operation on the teeth, such as grinding, 
promised to be unnecessary. By virtue of the basic molding generation 
process entailed, it was at once apparent that a gear rolling machine of pre- 
cision could be built which would produce gearing of extreme tooth accuracy 
in one operation and that such a machine would make it possible to avoid 
error or approximation in the form of the teeth produced. 

MASTER FORM SYSTEM OF GEARING 

Further development entailed — in addition to the refinement of the rolling 
machine necessary to convert it into a practical tool for the quantity pro- 
duction of gears — the adoption of a simple system of gearing which would meet 
the requirements of low cost, precision and ease in die-roll design and con- 
struction. Naturally, many tooth forms can be employed for the die-roll 
or master rolling gear, but it is quite essential to adopt a standard tooth 
section which can be reproduced with the utmost precision and at a minimum 
expense, if accuracy in the formation of the rolled teeth and low costs of pro- 
duction are to be realized. That is, the die-roll tooth form should be suscep- 
tible of accurate and economical reproduction. 

The surface which is reproduced with greatest precision and at the same 
time at minimum expense by mechanical means is the plane. Consequently, 
the adoption of a straight-sided tooth, the profile surfaces of which are 
perfectly flat, is the logical selection for die-roll teeth. Such tooth form is 
also distinctive of the basic involute rack and of the octoid crown gear, so 
may very properly be termed a "master form." It is eminently suitable, 
for the die-roll teeth, meeting the requirements for their design and construc- 
tion, and the teeth rolled by die rolls so proportioned resemble in appearance 
the familiar involute gear tooth. 

The rolled teeth differ slightly in profile curvature from the true involute, 
it is true, but as the gears of different size rolled by a common die roll have 
teeth which are conjugately related as to section this is quite immaterial. 
In fact, the effective length of tooth profile — the limits of the path of tooth 
contact — is somewhat greater than in the case of the involute system of gearing 
as adapted in commercial gear production, for a certain relief is provided at 
top and bottom of teeth by the straight die-roll tooth profile, without sacri- 
ficing the duration of tooth contact. 

The reason for this relief in the case of spur, helical and herringbone gears — 
all of which types can be rolled with equal facility — is readily understood, 
as the die-roll teeth resemble in section the teeth of an involute rack. That 
is, the thicknesses of the teeth at top and bottom are somewhat greater than 
if the teeth were of true involute form. In the cases of gears of the bevel 



ROLLED GEARING 



337 



form, the same relief is secured by making the die rolls in the form of very 
flat bevel gears, instead of true crown gears. The straight profile teeth are 
retained, as if the die rolls were of the true crown gear form, but the center 
angle of the die roll is made somewhat less than 90 degrees. The effect of 
this is the same as in the case of die rolls for spur gears, for a similar 
relationship exists between the straight Hne profile teeth of the flat bevel 
die roll and the teeth of a true crown gear as between the straight line profile 
teeth of the spur die roll and the teeth of an involute rack. In the case 
of the bevel gears, however, a true crown gear form of die roll can be used by 
making certain modifications in tooth proportions, as will later be explained, 
but this is not usually attempted. 




FIG. 263. FRONT AND SIDE VIEW OF BEVEL RING GEAR ROLLING MACHINE. 



PROCESS OF GEAR ROLLING 

The rolling, or forging, machine is shown in Figs. 263 and 264. It consists 

essentially of two independent shafts — one supporting the gear blank and 

the other the die roll — driven at synchronized speeds through substantial 

timing gears, and can best be described by a brief account of the actual process 

of gear roUing. A gear blank heated to its most suitable forging temperature 

— ordinarily in the neighborhood of 2,000 or 2,100 degrees Fahrenheit — is 

mounted on the work shaft and clamped securely in place as shown on the 

right-hand arbor. Fig. 265. The water-cooled die, carried on the other 

functioning shaft, on starting the machine is advanced by a smooth positive 

loam action and gradually forced — under a pressure of 10 to 20 tons — 
22 



33S 



AMERICAN MACHINIST GEAR BOOK 



into engagement with the semi-plastic gear blank, progressively displacing 
the metal and gradually building up the molded gear teeth. The work 
and die are kept in this rotational contact until the teeth are fully formed, 




FIG. 264. REAR AND SmE VIEW OF BEVEL RINo oEaK kuLLING MACHINE. 

the die roll meanwhile advancing to full mesh. The heated gear blank has 
in the meantime cooled to below its critical temperature, so that the formation 
of forging scale has also ceased. The die roll is then withdrawn and the 
rolled gear removed to cool. 




FIG. 265. DIE BLANK AND TIMING GEARS — BEVEL RING GEAR ROLLING MACHINE. 

As the rotary speeds of the die roll and gear blank are positively synchro- 
nized at the speeds of their engaging pitch surfaces through the heavy timing 
gears, there is no driving action between the die roll and gear blank and the 
teeth of the die roll are constrained to enter the blank on radial lines and in the 
same relative position on each successive revolution. The advancement of 



ROLLED GEARING 



339 



the die roll, with the accompanying displacement of gear blank metal, is 
slight per revolution, so that teeth on the gear blank are molded gradually 
and without strain. Throughout the process of rolling, the temperature of 
the die roll does not rise above that which is bearable to the hand, being kept 
cool by a stream of water directed against its face at the point farthest from 
that of its contact with the heated blank. 

When gears of extreme accuracy of tooth structure are required, the 
work and die-roll shafts are made to reverse their direction of rotation 
frequently during the greater part of the process of building up the teeth. 




FIG. 266. FIG. 267. 

FIG. 266. BEVEL RING GEAR AND PINION BLANKS. 

FIG. 267. ROLLED BEVEL RING GEAR AND PINION BLANKS. 

This reversal is performed at a speed of about 150 r. p.m. and it has the effect 
of balancing the displacement of metal on either side of the teeth, so forming 
a perfectly symmetrical tooth structure. The reversal in rotation, auto- 
matically performed, has heretofore been accomplished through the agency 
of powerful friction clutches, but in the latest types of machines special 
D. C. reversable motors have been substituted with a considerable saving in 
machine cost and in the over all dimensions of the rolling machine. 



ELIMINATION OF FORGING SCALE 

During the rolUng operation, the stream of cooling water directed against 
the die roll serves to wash free any forging scale which may tend to cling to 
the die-roll teeth and is also instrumental in ridding the gear blank of the 
scale as rapidly as it is formed. The cool, wetted die-roll teeth coming in 
contact with the hot gear blank accentuates its rate of shrinkage, loosening 
the forging scale as it forms. The speed of blank rotation then throws the 
scale free of the machine, leaving the surfaces of the rolled teeth entirely 
free of clinging scale. As rolling process is continued until the gear blank 



340 



AMERICAN MACHINIST GEAR BOOK 



cools below its critical temperature, the forged teeth are highly polished, 
free from any imbedded scale or other blemish when the formed blanks are 
removed from the rolling machine. 

FINISHING ROLLED GEARS 

In fact, the teeth of the gears are finished with the rolling operation and 
nothing remains to be done other than to bore and face the hubs and to dress 
the shrouding formed by the molding of the gear teeth. For these finishing 
operations, the rolled gear blanks are chucked on the accurately rolled pitch 
surfaces of the gears — to assure perfect centering — and the machining 
performed on suitable automatic machines of standard type. 




FIG. 268 FIG. 270. 

FIG. 268. FINISHED BEVEL RING GEAR AND PINION. 
FIG. 270. HERRINGBONE BEVEL GEAR AND PINION. 

The shrouding tying the formed teeth and gear body, or web, into an 
integral unit is due to the use of die teeth somewhat shorter than the face 
width of the gear blanks and may be entirely cut away, producing gears of 
the usual form, or may be retained in whole or in part to add strength to the 
gear. In the case of a pair of meshing gears, a full shroud or a part shroud 
may be retained on the member which is customarily the weaker of the two, 
making it the equal or superior of the other in strength. Or, shrouds on 
both members can be so proportioned as to develop the maximum or the 
most effective strength of the gear combination. 



INHERENT STRENGTH SUPERIORITY 



The heavy pressure under which the semi-plastic blank metal is worked 
into teeth during the rolling process is also instrumental in increasing the 
strength of rolled gearing by bringing about a marked rearrangment and 



ROLLED GEARING .^j 

modification of the metal structure. The plastic metal during the gradual 
molding of the teeth is subjected to a thorough and powerful kneading in 
process which breaks up the more or less heterogeneous crystalHne structure 
of the blank into one of much finer and uniform crystals arranged in a trussed 
formation of almost fibrous characteristic. This distinctive rearrangement 
of metal structure serves to tie the teeth to the body of the gear blank in a 
way which adds greatly to the strength of the teeth and serves also to equalize 
any warpage effects which might otherwise develop in subsequent heat 
treatment. The density of the tooth structure is increased, developing as 
well, teeth of superior toughness. 

Exacting laboratory tests, which have been duphcated or bettered under 
working conditions, have shown that the average superiority of rolled gear 
teeth-comparmg them with high grade cut teeth from similar metal blanks 
—IS a gam of 25 per cent, in strength and one of about 20 per cent in hard- 
ness ^ In operation, this superiority means greatly increased wear resisting 
quahties and a capacity to withstand successfully heavy and sudden 
overloads. 

PRODUCTION ECONOMIES 

As the process of gear rolling is essentially one for large quantity pro- 
duction, the resulting economies are probably of even greater importance 
than such vital considerations as accuracy of tooth formation, resistance to 
wear, hardness, toughness and strength of gear teeth, and in this connection 
the rolling process shows a number of marked advantages. Important 
savings are reahzed by the process in material, labor and equipment costs 
and also by the much smaller space required for the accommodation of the 
plant needed for a given production output of gears. 

The teeth being formed on the gear blanks by a molding process, rather 
than by the removal of any metal, as in all gear cutting processes, the gear 
blanks can be so proportioned that only a minimum amount of metal is 
trimmed away in finishing the gear hubs and the ends of the gear teeth 
This enables a saving to be made of from 20 to 40 per cent, in the weight of 
the rough blank. ^ 

The crew for operating a rolling machine and its supply furnace for 
heating the gear blanks consists of only two men, an operator and a helper 
who under mtelligent.direction from a competent foreman may be recruited 
from the class of intelligent laborers. Such a crew can easily attain and 
mamtam an average hourly output of 90 rolled gears per machine. Com- 
pared to the labor expense in simply cutting finished teeth by the most 
economical process of machining, at a rate of 90 gears per hour, the saving 
by the rolhng method amounts to between 90 and 97 per cent. Finishing up 
the gears somewhat cuts down the superiority of the rolling method, but the 
savmg IS still very marked, being from .5 to 80 per cent., or ev^n more 
depending upon the size and type of gearing produced. 



342 



AMERICAN MACHINIST GEAR BOOK 



The big output of a gear rolling machine — a lo-inch gear with 2 -inch face 
consuming only a trifle over 14 seconds in the rolling operation — results in 
considerably fewer of them being required for a given rate of production than 
the number of gear cutting machines needed for the same gear production. 
Consequently, a substantial saving is effected in necessary equipment invest- 
ment and a considerable saving in floor space. As for the investment saving, 
the case of a shop having a daily production capacity of 700 ring bevel gears, 
700 bevel pinions, 700 miter bevel gears and 1,400 spur gears will prove 
typical. 

The gear rolHng machines and heating furnaces for such an output would 
cost 75 per cent, less than the equivalent equipment in generating machines 
and machine tools. In the finishing up operations — the boring and facing 



FIG. 




CROSS-SECTION OF ROLLED GEAR TOOTH SHOWING STRUCTURE. 



of hubs, etc. — the additional equipment cost favors the machining process to 
some extent, but in the over all equipment cost, that for all machines entailed, 
the rolling process is the more economical by some 55 or 60 per cent. This 
comparison of equipment costs is based upon established costs of gear rolling 
machines employing friction clutch reversing, or oscillating, mechanism. 
The latest type of roUing machines employing direct connected reversing 
motors in place of the friction clutches will be very much less costly — a 
saving of 30 or 40 per cent, being realized by the substitution — making 
possible a substantially greater saving in necessary equipment investment. 

In addition to the foregoing distinctive economies of the rolling process, 
a substantial saving is realized in the space of floor area required for the 
gear rolling equipment, but even disregarding this latter item, which can 
quite evidently be very important, the other savings realized through rolling 
instead of machining gears are so substantial as to be startling. Quite aside 
from the fact that hot rolled gearing can be produced which is far superior 



ROLLED GEARING 343 

to the best quality of generated gearing in accuracy of tooth formation, 
finish and strength of teeth and in wearing qualities, production of gears by 
this method shows a saving of from 20 to 40 per cent, in material, anywhere 
from 25 to 80 per cent, in labor expense and from 55 to 60 per cent, in the 
fixed charges represented by investment outlay. 

DESIGN OF ROLLED GEARING 

Great as are these economic, physical and mechanical superiorities of 
rolled gearing, still other advantages distinguish the process which enhance 
its value in quantity gear production and substantially increase the scope of 
commercial gearing. The simple straight line profile teeth of the die-rolls 
introduce simplification in gear design and make commercially feasible 
certain types of gears heretofore beyond the scope of practical gear produc- 
tion. 

The question of gear design in rolled gearing resolves itself into a rela- 
tively simple matter of die-roll construction and proportions, based upon a 
few simple formulas and rules, applicable to spur and bevel gearing generally, 
whether the gear teeth are of the straight, helical or herringbone form. These 
formulas are appended and a brief explanation of the three general forms of 
die rolls — two for bevel gears and one for spur gears — will make plain not 
only the steps in the design of die rolls but the enlarged scope or field in gear 
mechanisms opened up by the rolling process. 

BEVEL GEAR DIE ROLLS 

As previously stated, there are two varieties of die rolls for producing 
bevel gears, the flat bevel die roll and the crown die roll. The former is 
nearly invariably employed for rolling bevel gears of so-called standard 
pitches. The reason for this is that the radius of the crown die roll is gov- 
erned by the cone distance of the gear to be rolled, when the gear blank is at 
its critical temperature, and this distance is rarely an even multiple of the 
circular pitch. Consequently, it becomes necessary to decrease the center 
angle of the die roll until its cone distance (effective) equals that of the rolled 
gear blank at its critical temperature, thus providing for a full complement 
of whole teeth for the die roll. The circumference of a circle having a radius 
equal to the cone distance of the gear to be rolled divided by the circular 
pitch of the gear gives the number of teeth for a crown die roll. In the case 
of a bevel die roll, the quotient is the number of die-roll teeth plus some frac- 
tion. To ascertain the pitch diameter of the flat bevel die roll, the fraction 
is dropped and the product of the whole number multiplied by the coefficient 
of expansion is divided by the diametral pitch. The center angle of the flat 
bevel die roll is then readily obtained, the necessary offset to the die roll 
shaft etc. (See formulas for bevel gear die rolls.) 



344 



AMERICAN MACHINIST GEAR BOOK 



The use of the flat bevel die roll is necessary in order to produce a rolled 
gear of established pitch and consequently of fixed diameter, but it is quite 
conceivable that many occasions might arise where gear proportions slightly 
smaller or larger might prove very much more desirable than proportions 
established definitely by a standard of pitch. That is, gearing controlled 
by considerations of available space or by diameter offers a wider scope 
than a system in which the diameters of the gears are fixed by the 
pitch. In any production process, this would entail some modification in 




FIG. 271. MISCELLANEOUS GEARS MADE BY THE ROLLING PROCESS. 

pitch, but in the gear rolling system this is accomplished with the utmost 
precision by the use of a crown die roll having an effective diameter equal to 
double the cone distance of the gearing required when at its critical tem- 
perature. In the case of any gear cutting process, on the other hand, it 
would entail the use of special and costly cutting tools which could not be 
ground or reproduced with the same precision. The greater strength of 
rolled gearing, furthermore, affords somewhat greater leeway in the matter of 
pitch than may be taken with cut gearing. 

The redressing, or grinding, of bevel gear die rolls — an operation which 
may be necessary after rolling from 500 to 1,000 gears — is a simple proposi- 
tion, as all proportions of die-roll teeth are kept constant, but in the case of 
die rolls for spur gears the situation is somewhat different. The spur die roll 
is in the form of a gear with straight line profile teeth, which if redressed will 
reduce to some extent the diameter of the die roll and consequently the 
circular pitch of its teeth. Although this is true, it is possible to redress such 
die rolls until there is quite a measurable reduction in their pitch diameters 



ROLLED GEARING 



345 



without destroying the accuracy and smooth running qualities of the gears 

they can roll. 

This peculiarity is made possible by the positive driving of the two func- 
tioning shafts through heavy timing gears. As these timing gears control 
the angular advance of the die rolls and of the gear blanks, the effect of a 



p - Diarnehal Piich 
p'^ Circular Piich 
g - Pressure Angle 
h - Diameter Apex 
Circle -Helical 
'^' and Herringbone 
^ Gears. 

m=Parii'ng Diameter 
Herringbone Gears 




Bevel Die Roll 



Crown Die Roll 



FIG. 272. 



variation in the respective diameters of die and blank — within reasonable 
limits — is the introduction of a slight creep between the die and blank as they 
are brought into and run in synchronized contact, the width of the tooth 
spaces of the die-roll and not the thickness of the die-roll teeth governing 
the thickness of the teeth rolled on the plastic gear blanks. Consequently, as 
the radial spacing of the formed teeth is controlled and kept constant by the 
timing gears, it is essential that the die tooth spaces be maintained constant. 
If this is done, a slight reduction in the thickness of the die-roll teeth, due to 
the reduction in die roll diameter through tooth redressing, will not appreci- 



346 



AMERICAN MACHINIST GEAR BOOK 



ably modify the profile curvature of the rolled teeth and will not affect the 
accuracy of tooth spacing on the rolled gear. By maintaining constant the 
tooth spaces of spur gear die rolls, they can be redressed several times before 
they should be discarded, making the life of such dies measurable by a pro- 
duction of several thousand gears. 

HELICAL AND HERRINGBONE DIE ROLLS 

The simple straight line profile tooth adopted for die rolls being equally 
suitable for axial and helical, or spiral, arrangment of teeth, die rolls for 
heUcal gears, both of the spur and bevel variety, can be made and kept in 



p:= Diametral Piich 
p'- Circular Pitch 
g= Pressure yAngle j 
q=Spiral Angle ; 

^ (Helical a nc/ f 

Herringbone Gears) '' 




HeVical Die Roll 





y/^ uieToQ-fh Details 



d' Reduced kj 
, dressings 



Hernn q borie 
Die Roll 



FIG. 273. 



condition with almost the same ease as similar die rolls with straight axial 
teeth. This enables heUcal gearing to be rolled as cheaply as that with 
ordinary gear teeth. By making the die rolls in concentric sections, separable 
for tooth dressing, but otherwise keyed into an integral unit, herringbone 
varieties of either spur or bevel gearing with rigid binding tooth prows can 
be rolled just as cheaply. Herringbone bevel gears, which exemplify the 
scope of rolled gearing, cannot be produced by any practical process of gear 
cutting. They will doubtless prove of even greater commercial value 
than the herringbone gear type of spur gear, as they will eliminate the 
objectionable side thrust in most installations of bevel gearing. 



ROLLED GEARING 



347 



DIE ROLL FORMULAS AND DESIGN 



NOTATION BEVEL GEARING 



DIE ROLL 



Z — Coefficient of expansion 
D' = Effective pitch diameter 

N = Number of teeth 

E = Center angle 

F = Face 
F' = Face angle 
K = Addendum angle 

/ = Dedendum angle 

D = Outer diameter 
D" = Inner angle 

S = Dedendum 

C = Clearance 

G = Pressure angle 

X = Bevel angle (outer) 

Y = Bevel angle (inner) 

T = Width of tooth space 
T' = Normal width of tooth space 
W = Face cone distance 

Q = Spiral angle 

H = Diameter of apex circle 

M = Parting diameter (herringbone gears) 

T" = Normal width of tooth space (parting circle) 

O = Offset angle 

FORMULAS FOR BEVEL GEAR PIE ROLLS 



P 

P' 

d' 

d" 

f 
f 

s 
c 
e 
b 

J 
k 

g 
a 

V 

h 

w 

<1 
m 



ROLLED GEAR 

Diametral pitch 
Circular pitch 
Pitch diameter (outer) 
Pitch diameter (inner) 
Face 

Profected face 
Addendum 
Clearance 
Center angle 
Back angle 
Addendum angle 
Dedendum angle 
Pressure angle 
Pitch cone distance 
Root diameter 
Diameter apex angle 
Root cone distance 
Spiral angle 
Parting diameter 



Z = (i + temperature range) 0.00000672 



BEVEL DIE ROLLS 



N + fraction 



6.2832a 



(N = largest whole number) 
P 



Sin. E = 



0.5N 



ap 

= 90° - £ 
F' = E + K 

V = d' — 2{s -\- c) sin. b 



w = 



o.sv 



Sin. (e — k) 
W = Zw + 0.0625 
D = 2Z COS. (90° - F') 
D" = D - 2F COS. (90° - F') 



CROWN DIE ROLLS 

6.2832a 

P 

D' = 6. 2832^7 
D = D' + 0.125 

E — go deg. 

= deg. 
F' ^ go° + K 

Cos. K 
D" = D - 2F COS. K 



S + C = 



D(s + c) 



d' 



ALL BEVEL GEAR DIE ROLLS 

^ _Di 
^ ~ d' 



F = Z/+ 0.125 



T = 2W sin. 



180 

N 



h = d' sin. q 



X and Y are arbitrary within limits 

T' = T COS. Q 



HELICAL BEVEL GEAR DIE ROLLS 
H ^ Zh 



348 



AMERICAN MACHINIST GEAR BOOK 



H = Zh 



HERRINGBONE BEVEL GEAR DIE ROLLS 
M = ZVid'Y - {d' + d")f' 
T = T COS. Q . 
NOTATION FOR SPUR GEAR DIE ROLLS 



'T'// 



T'M 
D 



DIE ROLL 

Z = Coefficient of expansion 
D' = Effective pitch diameter 
N = Number of teeth 
F = Face 
5 = Dedendum 
C = Clearance 
T = Width of tooth space 
U = Bevel angle 
Q = Spiral angle 
T' = Normal width of tooth space 



ROLLED GEAR 

p = Diametral pitch 
/>' = Circular pitch 
d' = Pitch diameter 
/ = Face 
s = Addendum 
c = Clearance 
g = Pressure angle 
i = Advance of gear 
q = Spiral angle 



FORMULAS FOR SPUR GEAR DIE ROLLS 

Z = (i + temperature range) 0.00000672 

7N 

D' = ^ F = Z/ + 0.125 

S = o.siSsp'Z S -\- C = o.s6Ssp'Z 

i = i.ip' (Helical and herringbone gears) 



N = 24P + I 



Tan. G = -(Helical gears) 
U is arbitrary within limits 



2t 



Tan.G = -7 (Herringbone gears) 



INDEX 



Addendum and dedendum angles, 138-141 
Anderson system of rolled gears, 19, 334- 

348 
Anglemeter, 90-91 

tests, 91-98 
Arc of action, 26 
Arms, hollow, 117 

I-shaped, 109 

number, 106 

spur gear, 112 
Automobile worm drives, 181 

cost, 182 



B 



Back-gears, differential, 266-267 
Bevel gears, 130-151 

addendum and dedendum angles, 138- 
141 

design, 118 

diameters and angles, 142 

efi&ciency, 149 

formulas, 133, 134-136 

generating, 143 

grinding, 151 

hardening, 150 

herringbone, 208 

intermittent, 238 

machining, 143 

notation, 133, 134, 136 

Pentz parallel depth, 144-145 

special, 303-319 

spiral type, 303-316 

strength, 54 

"Williams "Master Form," 316-319 
Bilgram generating machines, 8-9 
Bore of gears, 125 "• 

Brown & Sharpe tooth standards, 20 
Buttressed tooth gears, 26 



Chordal pitch, 36-37 

thickness of teeth, 37-40 
Circular herringbone gear, 208 



Circular pitch table, 31 
Classification of gears, 2 
Connecting rod arm, 109 
Contact, paths of, 323-325 
Cored teeth, 103 

Cost of automobile worm drives, 182 
Cutters, involute, 41 
Cycloidal tooth, 3 



D 



De Laval speed reduction gear, 195 
Design, details, 102-129 

errors in, 91-96 

essentials of, 77 
Diametral pitch table, 30 

pitch worms, 188 
Differential back-gears, 266-267 



E 



Efficiency, 98-101 

bevel gear, 149 

herringbone, 207 

involute, 98 

large gears, 99 

lubrication, 95 

worm gear, 168 
Ellipse, Gardiners', 250 
Elliptical gears, 246-252 

center bore, 246 

cutting, 247 

fixture, 247 

interference, 252 

laying out, 246 
Epicyclic gear trains, 253-267 

analysis, 253-266 

compound, 257-259 

differential back-gears, 266-267 

direction of rotation, 260 

examples, 259 

formulas, 253-259 

internal, 256, 257, 258 

simple, 253-257 

velocity ratio, 258 
Errors in cast gear teeth, 91-96 



349 



350 



INDEX 



Face width, io6 
Factor of safety, 59 
Fellows' gear shaper, 9 

tooth standards, 21 
Fits, force, 125-129 
Friction driven lathe, 285-286 
drives, 268-302 
gears, 268-302 

coefficient of friction, 273, 278, 279, 280 
efficiency, 288-295 
endurance tests, 296-299 
fiber wheels, 273-277, 278 
conclusions, 277 
leather and aluminum, 274, 276 
and iron, 274, 275 
and type metal, 274, 276 
physical properties, 295 
resistance to crushing, 276 
straw and aluminum, 273 
and iron, 274 
and type metal, 273 
sulphite and aluminum, 276 
and iron, 276 
and type metal, 276 
tarred and aluminum, 275 
and iron, 274 
and type metal, 275 
metal wheels, 277 
power capacity, 279 
working pressures, 279 



Gardener's ellipse, 250 
Gear proportions, 102-129 

rolling process, 337-340 

shaper, Fellows', 9 

systems, Anderson, 19 

trains, arrangement of, 47-48 
epicyclic, 261-266 
Williams internal, 320-333 
Generating bevel gears, 143 

gear teeth, 3, 8 

machine, Bilgram, 8-9 
Geneva stop, 238-242 
Globoid gear, 183-188 

hob, 186 

machining, 185, 187 

strength, 184 

worm, 185 
wheel, 187 



Grant's tooth standards, 20 
Grinding bevel gears, 151 

H 

Hardening bevel gears, 150 

Hardness of gear teeth, 77 

Helical gears, examples in design, 193-195 

Helical and herringbone gears, 190-209 

Herringbone gears, 196-209 

advantages, 202 

angle of teeth, 205-207 

applications, 203-205 

bevel, 209 

circular, 208 

efficiency, 207 

interchangeable system, 197 

lead, 205-207 

machining, 198-202 
hobbing, 199 
milling, 201 
planing, 201 

modified, 208 

strength, 207 
High speed gearing, 84-90 

material for, 84 
Hindi ey worm gear, 183-188 

hob, 186 

machining, 185, 187 

strength, 184 

worm, 185 
wheel, 187 
Hobbing herringbone gears, 199 
Hobs, 157-165 

double threaded, 163 

flutes, 158 

Hindley worm, 186 

length, 157 

relieving, 164 

single thread, 162 

spacing, 164 

spiral fluted, 164 
Hoist gearing, example in, 49-52 
Hot rofled gears, 77, 334-340 
Hunt's tooth standards, 21 
Hunting tooth, 28 



Interference, elliptical gear, 252 
internal gear, 320 
involute, 11-14 
rack and pinion, 14-15 



INDEX 



351 



Intermittent gears, 236-245 
bevel, 238 

Geneva stop, 238-242 
spiral, 243-245 
spur, 242 
worm, 243 
Internal gearing, limitations of involute, 

320 
Williams system, 320-^^^ 

advantages, 332-333 

costs, 332 

formulas, 321 

machining, 321 

reduction in number of pinion 
teeth, 330 

strength of pinion teeth, 327 

wear, 327 
Involute, the, 5 
cutters, 41 
drawing curve, 9-1 1 
teeth, 6-15 

action, 7 

generation, 3 

interference, 11- 15 
I-shaped arms, 109 



K 



Keys, Kennedy, 124 
square, 122 
tapered, 122 
Woodruff, 124 

Keyseats, 122-125 



Lanchester worm gear, 181 
Lead, helical gear, 192, 193 

herringbone gear, 205-207 

spiral gear, 216 
Lewis formulas, 53-58 

modified, 54, 57 
Life of gears, 76 

Limitations, involute internal gearing, 320 
Line of action, 25 
Logue's tooth standards, 21 
Lubrication, 76, 95 

M 

Machining bevel gears 143 
herringbone gears, 196-202 
skew bevel gears, 233-235 



Metric pitch, 35-36 
Milling bevel gears, 143 
herringbone gears, 201 
Miter gears, 133-135 
Modified involute teeth, 23 
Molding process, 8 
Mortise gears, 121 

O 

Octoid tooth generation, 3, 24 

standards, 23 
Origin of involute teeth, 6 



Paths of point of contact, 323-325 
Pentz parallel depth gears, 144-145 
Pitch, chordal, 36-37 

circular, 31 

definitions, 29 

diameters, 35 
formulas, 42-45 

diametral, 30 

metric, 35-36 

normal, spiral gear, 212 
Planing herringbone gears, 201 
Power ratio, 48 
Proportions and design, 102-129 

arms, 104, 109, 11 2-1 17 

bevel gears, 118 

bore, 125 

connecting-rod arm, 109 

cored teeth, 103 

face width, 106 

formulas, 103-104 

hollow arms, 116 

hub diameters, outside, 104 

I-shaped arms, 109 

key, Kennedy, 124 
Woodruff, 124 

keyseats, 122-125 

mortise gears, 121 

rawhide gears, 1 18-120 

rim gears, 117 

skew bevel gears, 228-229, 233-235 

split gears, 107-109 

spur gears, 102 

webbed gears, 106 

R 

Rack and pinion interference, 14-15 
Railway gears, 52 



352 



INDEX 



Ratio, power, 48 

speed, 46 
Rawhide gears, 1 16-120 
Resistance to wear, 88 
Reversible worm and gear, 156 
Rim gear proportions, 117 
Rolled gearing, 334-346 

design, 343 

development, 335 

die rolls, 343-346 

economies, 341-343 

finishing, 340 

forging scale, 337 

formulas, 347-348 

shrouded, 340 
Rolled gears, Anderson system, ig 

proper design, 77 

speed, 81 

strength and hardness, 77-78 

superiority, 81 

teeth, 335 

master form 336 
notation, 347, 348 
process, 337-340 
strength, 340 



Safety, factor of, 59 
Sellers' tooth standards, 21 
Shrouded gears, 60-65 
Skew bevel gears, 228-235 

examples in design, 232-233 

formulas, 230-232 

machining, 233-235 

notation, 230 
Speed, highest recorded, 85 

rolled gear, 81 

ratio, 46 

reducer, De Laval, 195 
Speeds, 80-81 

influence of location, 82 
pressure, 82 
Spiral gears, 210-227 

cutters, 227 

examples, 216-220 

formulas, 217 

helix, 212-214 

intermittent, 242 

laying out, 220-222 

normal pitch, 212 

notation, 216 

rotation, 225-227 



Spiral gears, speed ratio, 214-215 
tables, 223, 224 
thrust, 225 

type bevel gears, 303-316 
cutter standards, 313 
design, 310-316 

comparison of methods, 315-316 
formula method, 310-313 
layout method, 314-315 
face, 307 

formulas, 31&-313 
machining, 308-310 
notation, 310, 311 
pitches, 306 
pressures, 304 
spiral angle, 306 
thrust, 304-305 
tooth proportions, 308 
Spur gear calculations, 42-45 
formulas, 43-45 
notation, 42 
Spur gears, mortise, 121 
rolled, 334 
shrouded, 60-65 
speed, 79 
split, 107 
webbed, 106 
Steel for gears, 87-88 
Stepped gears, 28 
Strength, 53-58, 78 
herringbone gears, 207 
Hindley worm gears, 184 
rolled gears, 77-78, 340 



Teeth, comparative sizes, 32-34 

shrouded, 65 

wear, 65 
Templets, gear tooth, 28-29 
Thickness, chordal, 37-40 
Thrust, spiral type bevel gears, 304-305 
Tooth, buttressed, 26 

contact, 25, 80 

cycloidal, 3 

generation, 8 

involute, 3 

modern developments, 15-17 

octoid, 3, 24 

parts, 2-3 
Tooth standards, 20-24 

Brown & Sharpe, 20 

Fellows', 21 



INDEX 



353 



Tooth standards, Grant's, 20 
Hunt's, 21 
Logue's, 21 
octoid, 23 
Sellers', 21 
Universal, 23 

U 

Universal tooth standards, 23 

V 

Velocity ratio, epicyclic gear train, 258 

W 

Waterbury watch movement, 264 
Wear, resistance to, 88 
Webbed spur gears, 106-107 
Weight, 110-112 

Wilhams "Master Form" bevel gearing, 
316-319 



Williams, economies, 318 

Williams system of internal gearing, 320- 

333 
Worm and gear, reversible, 156 
Worm drives for automobiles, 181 
Worm gears, 152-189 

cost, 182 

efficiency, 168 

examples, 174-179 

formulas, 154-157 

globoid, 183-188 

Hindley, 183-188 

Lanchester, 181 

manufacturing processes, 166-167 

materials, 167 

notation, 153 

pitch diameter, 165 

power, 168 

pressures, limiting, 179 

speed, limiting, 179 

straight cut, 167 

temperature, 180 



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